It is barely breakfast time on a bleak Brussels morning but 230 people have squeezed into the conference room of a city-centre hotel to see the closest thing there is to a rock star in the nerdy world of technocrats and international finance.

Christine Lagarde, the head of the International Monetary Fund, is launching a book by IMF economists on the rather dry topic of European job creation.

But Lagarde, the only person to have been on the cover of Forbes at the same time as being featured in Vogue , would probably have filled the hall with journalists, bureaucrats and think-tank analysts if she had been launching the latest edition of the Belgian phone book.

Germany’s Wolfgang Schäuble, the most powerful finance minister in Europe, has turned up on this dreary morning as a favour to Lagarde, and yet it is the tall, silver-haired woman in a grey Chanel outfit who attracts the TV cameras and who has middle-aged economists smiling like teenagers into their mobile phones as they take “selfies” standing near her.

Thirty minutes into the panel discussion on economic reform, a journalist from Le Figaro raises the question on everybody’s mind. Given that the topic was European jobs, he says with a knowing look, could Ms Lagarde imagine taking “a big job in Europe”?

“I will leave [your] question because it is not our focus, it is not my focus,” she replies crisply, as smiles and nods roll around the room.

At the end of the session, as Lagarde is ushered from the room with her mobile phone already held firmly to her ear, her audience is still buzzing with the observation that she chose to side-step rather than flatly reject the notion that she might leave her job running the world’s lender of last resort, where she oversees a trillion euros of loan capacity, to become the new public face of Europe.

Fabian Zuleeg, chief of the European Policy Centre, a Brussels-based think tank, and the host of the event, says there is “enormous interest right across Europe” in whether Lagarde will soon replace the lacklustre Portuguese José Manuel Barroso as head of the European Commission, the executive arm of the European Union. “Schäuble has already said that he would like to see her in a big European job,” Zuleeg says, “and she is the only person that there is any real excitement about.”

More than one government across the continent sees the charismatic 58-year-old Frenchwoman as the person best equipped to bring more cohesion to the European Union, and perhaps even start to bridge its vast political gulf with a half-billion population that often shows little interest or respect for its work.

Others believe that Lagarde, the former Finance Minister of France, could one day end up as that nation’s President.

When I met Lagarde the next day in a plush suite at the IMF’s offices in Paris I asked her about the conjecture that she might end up in the European Commission job when it becomes vacant in October, even though she would then have almost two years to run in her IMF term.

“I want to complete my five-year term,” she responds firmly. “I am not campaigning, I am not a candidate, I am not planning to leave.” But expressing her current intention surely is different to declaring that she definitely would stay?

“What about if I get run over by a bus, God forbid? Destiny is not in my hands.”

Lagarde’s remarkable career owes little to destiny. It is a product of ferocious hard work and a refusal to be deterred by the prospect of being the only woman at a table full of men.

Building her career without the support of a well-connected partner such as Bill Clinton or a wealthy patron such as Denis Thatcher, Lagarde was the first woman to lead a global law firm (in 1999), to be the finance minister of a major industrial country (2007), and to run the IMF (2011).

Being a woman at the top of law, politics and finance has meant a life of being “well surrounded but quite lonely”, she says.

“It is very distressing that there is in many corners still hostility towards women entering certain fields. I don’t think we have to be aggressive about it or threaten anybody, but it is clearly the case that men and women should be properly represented, should be able to access fields, circles and areas where they have perfect competence to participate, to lead and to be respected.”

Lagarde joined the Chicago-based law firm Baker & McKenzie at the age of 25 after a French firm had told her that it might give her a job but it would never make her a partner because of her sex.

At 39 she was promoted to the firm’s global executive committee, meaning she had to spend half her time in Chicago. Her sons Pierre-Henri, then nine, and Thomas, then seven, stayed in Paris with their banker father Wilfried Lagarde, from whom she had recently divorced.

Four years later she became the firm’s president and moved to Chicago, so throughout her sons’ teenage years she was in Paris for just one week a month.

In 2011 she told an interviewer that “I had to accept I could not be successful at everything, you draw up priorities and you accept a lot of guilt”.

To me, she notes that men in similar situations come under much less pressure.“I don’t think you would ask me this question if I were a male, that is true .”

“I wish some of them would feel a bit more guilty about [making family sacrifices] as well, but I am not sure they do. But that sense of guilt fades away over time. As you age, it reduces because children grow up, grandchildren arrive and you sort of reconcile yourself with what you have done.”

Friends say she has excellent relationships with her sons, but also that, as one puts it, “she is trying now to make up a bit for lost time with the boys”.

Adds Lagarde: “My companion [Xavier Giocanti] has two children whom I very, very much love and one of them has a little boy and is expecting a baby girl, so I regard myself also as a grandmother.”

Lagarde has for many years spoken out on women’s issues, backing executive quotas for women, favouring them for promotion within the IMF and chiding countries like Japan for not doing enough to involve women in their workforce.

Susan Schwab, George W. Bush’s former trade representative, has known Lagarde since she was France’s trade minister from 2005 to 2007.

She says Lagarde has had to be careful in how she advocates women’s issues. “She has been conscious not to do it in a way that pigeonholes her or undermines the perception of her as a well-rounded professional.”

Another old friend, Sir Peter Westmacott, now the British Ambassador to the US, says Lagarde is as passionate about women’s issues in private as she is in public.

“I have seen her raise those matters in big speeches but I have also seen her do it in tiny groups by encouraging women not to be reticent in fulfilling their ambitions.”

While Lagarde has never been able to take advantage of the male networking that boosts many men’s careers, she has turned her own busy lifestyle into an asset.

One of her key strengths in endless rounds of late night talks is her durability and stamina. A former synchronised swimming champion, she does 20 minutes of yoga each morning, swims when she can, rarely drinks alcohol and has no meat, coffee or tobacco.

Nowadays, she needs six hours of sleep instead of the five hours that have sustained her previously, but friends say she is still generally the last person standing in late-night negotiations, when she is often up against middle-aged, slightly overweight men.

Lagarde’s biographer Cyrille Lachèvre, a former business editor of Le Figaro , says that beneath her charm is the competitive streak of an athlete. He recalls that during one long round of talks in Brussels over German attempts to stop France reducing restaurant taxes,Lagarde ensured that no food was served to the negotiators “and the Germans eventually gave up out of hunger”. Asked about that incident, Lagarde gives a conspiratorial smile. “We were going nowhere, so I told the chairman of the meeting, ‘If you want to wrap it up, you have to kick out of the room all the advisors so the ministers have to make a decision themselves … and don’t serve any food’. ”

Another story has it that Lagarde sometimes wears down her male rivals by draping herself warmly and organising extra air-conditioning.

To me she’ll concede only that she always equips herself with two items that men lack, “a scarf or a shawl, something to keep my shoulders warm or my neck warm, and a Spanish or Japanese fan, depending on the colours of my clothes”.

Clothes are clearly important to her; on the day we meet she is wearing a jacket made to her own Chanel-inspired design from material she gave a Paris couturier.

When Lagarde first entered the French ministry there was some media criticism of her eye-catching jewellery and taste for expensive Hermès handbags. In one incident her political “handlers” even organised for expensive jewellery to be airbrushed out of a photograph of Lagarde, but she quickly rejected the attempts to tone down her taste for fashion.

Given that she now earns a tax-free salary of more than $480,000 (€352,345), Lagarde can certainly afford her own handbags and jewellery. The international flavour of Lagarde’s career path owes much to the influx of Americans to Paris just before and after World War II.

“I had very cultured, intellectual grandparents on my father’s side and they had many American friends,” she says.

“Jazz was often heard in my father’s family. My grandfather was a painter and he was associated with American painters visiting here.”

Her father became a professor of English literature and she grew up in a household with a strong interest in “Anglo-Saxon” culture. At 17, she spent a year as an exchange student at the same posh girls’ school in Washington’s suburbs that Jacqueline Kennedy Onassis attended, honing her English and interning with a Republican congressman.

Back in Paris, she was twice rejected by the École Nationale D’Administration (ENA), the elite training school for politicians and bureaucrats, before studying law and then choosing the world’s largest international law firm, Baker & McKenzie, over French firms.

Sonia Criseo, her Irish-born personal assistant at the law firm and later in government, says Lagarde has always been an outsider. “She was a French teenager in America, then a European at Baker & McKenzie,” she says. “She was in government without being a graduate of ENA or a civil servant. She’s generally been a woman among men, of course, and even now she is running the IMF as a lawyer among economists.”

“I think being an outsider has actually been a strength for Christine because she’s been able to take a totally outside view of whatever domain she is in,” says Criseo, who remained her closest aide for 18 years before taking a private sector job when Lagarde went to the IMF.

“She has a discipline and capacity for work I have never seen before. She has huge concentration. As a working mother, she would try to go home for dinner and then come back to the office, or work late into the night at home.”

When her countryman Dominique Strauss-Kahn was forced to step down as IMF chief in 2011 because of an alleged sexual assault on a New York hotel chambermaid, Lagarde was seen as the ideal candidate to help Europe retain its 70-year hold on the IMF’s top job.

But Strauss-Kahn’s disgrace was not the first time that Lagarde was what Criseo calls “the right person in the right place at the right time”.

Her elevation in 1999 to run Baker & McKenzie came when it desperately needed to change its culture and image.

In the previous few years, the firm had been successfully sued by a secretary for sexual harassment, by a female partner for sex discrimination and by a male employee for discriminating against him for having AIDS, a case that inspired the 1993 film Philadelphia .

Cyrille Lachèvre says Lagarde later made a name for herself beyond legal circles at the World Economic Forum in Davos, Switzerland, recalling that by the year 2000 “people like [advertising executive] Maurice Lévy began asking, ‘Who is that smart French woman with the fantastic English?’ ”

Five years later, the conservative French president Jacques Chirac appointed her as a junior minister for trade.

The glamorous lawyer’s arrival ruffled the feathers of Paris’s career politicians and bureaucrats, who resented her foreign ways enough to dub her “la Americain”. Lagarde has, for instance, always preferred to work in English, a choice that is mildly scandalous in France.

Her time in the US had also left her more market-oriented than most of her compatriots, and deeply sceptical about French work practices such as the 35-hour week.

But Lagarde was never ideologically committed to the conservative cause nor even a reliable conservative voter: she voted for Socialist President Francois Mitterand in 1981.

Her diffidence about party politics has left her without a factional power base in Paris. “I am fascinated by politics but I am not a political animal,” says Lagarde, who has never been elected to a public office.

“I have strong principles and beliefs, about the strength of the individual and the sort of old-fashioned, 18th-century economic liberalism. I am a very strong believer in free access for all, provided that they put the effort in, so I have many principles. Does that put me on the right, on the left, in the centre or in a political party? I don’t think so.”

During a ministerial visit to Marseille in 2006, Lagarde met Giocanti, an old university colleague, and they have been an item ever since. The extroverted Marseille businessman has joked that he is Lagarde’s “Gross Domestic Pleasure” but her relentless travel means they are together only one week a month.

When Lagarde comes to private dinners or embassy functions in Washington, says Peter Westmacott, “Xavier is sometimes with her but she is usually on her own.”

In 2007, Lagarde was moved to the agriculture portfolio for a few weeks before the new president, Nicolas Sarkozy, gave her the crucial post of finance minister.

Cyrille Lachèvre says she made little progress in trying to reform policies like the 35-hour week “and by the beginning of 2008 we were all thinking that she would not last much longer as a minister”.

It was the collapse of US investment bank Lehman Brothers in September of that year and the global financial crisis that, in Lachèvre’s words, “saved her ministerial career” and thrust Lagarde into the spotlight as one of the key architects of the later efforts to save the euro.

“Sarkozy discovered that George W Bush and his team really liked her, and in Eurozone negotiations she was incredible,” says Lachèvre.“[German Chancellor] Angela Merkel and Schäuble came to trust her more than they trusted Sarkozy because she operated like a lawyer with a brief rather than a politician; when she said something it would stand.

“She has a mandate from her client, whether it is the president or the IMF board, and she has to do everything to argue that case and find a deal. You never know if she personally agrees with what she has been asked to pursue.”

I put that critique to Lagarde, mentioning the contradiction between her advocacy at the IMF for free trade and her record as a French minister defending the bloated farm subsidies of the European Union’s Common Agriculture Policy (Cap).

“The Cap was something I was asked to defend,” she replies, “and I am personally convinced it was a good cause in many respects – not all respects but in many respects.”

She says the IMF is now broadening its focus from the drier and more technical aspects of international finance to place new emphasis on issues such as green growth, income inequality and gender inclusion.

“We have found out, based on our research, that sustainable growth is better, and to actually feed sustainability, better [gender] inclusion and less [income] inequality are highly recommended.

And obviously green growth, so to speak, is an area that has to be explored, developed and endorsed by as many as possible.”

Lagarde agrees that she has personally taken the lead role in broadening the fund’s focus: “I am also passionate about drier things – making sure the financial sector is properly regulated, properly supervised, that we do not let the excesses and the abuses that we saw in the early 2000s be repeated.

“[But] on issues such as inclusion, reduction of inequalities and greener growth, I am personally passionate.”

Any move to draft her into the European Commission later this year would face two political hurdles. One is she would have to be nominated by France’s Socialist President Francois Hollande – and he may not want to promote a conservative rival.

Just as important is the fear that if Lagarde left the IMF, Europe would lose its top job, just when the continent is relying on the IMF for billions of dollars in funding.

In 2011, when the UK first pushed her candidacy for the IMF, Australia, Canada and Spain came out in support of a Mexican rival, Agustín Carstens, arguing that it was time for Europe to give up the job.

Former IMF chief economist Simon Johnson argues that Europe’s dominance must end. “It is not supposed to be the European Monetary Fund, and she has been very favourable to the Europeans during the whole euro crisis,” he says.

Domenico Lombardi, a former Italian member of the IMF board, agrees, claiming that at the height of the Greek debt crisis Lagarde’s IMF tried too hard to protect the largely French and German banks that had lent money to Athens.

Lagarde denies that she has been soft on Europe, insisting that Greece and other countries have been pushed to make even larger fiscal turnarounds than those demanded of Asian countries in the 1990s.“I have always tried to not be French, not be European, when I do my job,” she says. “And I think I have achieved that.”