The hardline, tough-as-nails image, which has served Margrethe Vestager well in Brussels while taking on multi-billion euro multi-national corporations such as Apple, comes with a price tag back in Denmark.

"She is respected for her work, but she doesn't have many friends at home. Particularly not amongst the Social Democrats," according to Marlene Wind, professor at the Department of Political Sciences, University of Copenhagen.

Student or retired? Then this plan is for you.

Danish PM Lars Lokke Rasmussen has already endorsed Vestager (Photo: Venstre)

The 'steeliness' of the current commissioner for competition, which recently inspired the Economist to publish a gleaming endorsement of Vestager, explaining why "she ticks all boxes" as the future president of the European Commission, has left her with a string of enemies at home - making her an unpopular choice at least for some.

On Tuesday (28 May), the Danish prime minister and leader of the Liberal Party, Lars Lokke Rasmussen, endorsed Vestager as his favoured candidate for the post as EU's next Commission president.

But Mette Frederiksen, leader of the Social Democrats - and likely to become the next prime minister - has been keeping silent: handing a top position to a rival liberal before the elections would not go down well within his party ranks.

"She has to [keep quiet]. Pointing to Vestager now would be regarded a 'sell-out'," explained professor Wind.

The questions is whether Vestager has made such an impression in Brussels that she doesn't actually need much support from home.

"No [Danish] prime minister would say no, if other EU leaders were to suggest a Dane for this position. One caveat could be the fact that Lokke Rasmussen has only recently endorsed [her], and that lacking a strong sponsor, she might lose her shot at the presidency.

"Personally, I believe her strong profile makes a long explicit campaign less important as long as it is known among the heads- of state and government that she is supported for the commission top-job. In addition, [EU Council president Donald] Tusk has just pointed to the need for gender balance, which is a very strong hand to Vestager," said Wind.

But for Rasmussen, Vestager and her Social Liberal Party, an ALDE ally, is a natural choice.

Danish People Party bleeding voters

In the southern part of Copenhagen, two shiny black sedans are parked outside a red brick apartment block. Here, a basement office is serving as the modest headquarters for far-right party, Stram Kurs - or The Hard Line Party.

Led by Rasmus Paludan, a 37-year old lawyer, who was recently convicted of racism, Stram Kurs has passed the threshold for voter support to stand in Denmark's parliamentary elections on June 5.

Calling for the deportation of all Muslims and regularly publicising his demands through provocative anti-Islam demonstrations, Paludan is under constant watch by police intelligence officers.

Lately, the nationalist far-right Danish Peoples Party (DPP) has been bleeding voters, losing three out its four seats in the European Parliament last weekend.

But make no mistake, says Wind, anti-immigrant and anti-Muslim voters have not disappeared - but are merely being scooped up by other parties such as Stram Kurs and the mainstream parties.

"No matter how many times the media publishes the fact that asylum seekers are down by 90 percent since 2015, politicians quickly revert to the familiar agenda of 'outside pressure on European borders' and 'terror threats'," she said.

"Facts, it seems, are not carrying much weight at the moment. Now, these new far-right parties are appearing, pushing mainstream parties towards an increasingly harder line on immigration," said Wind.

Denmark recently extended its "temporary" border control, which was introduced in 2016 after large numbers of refugees from Syria entered Europe, for another six months.

"The 'EU hasn't been up to the task'. However incorrect, this seems to be a widely-accepted fact, which both the Social Democrats and the Liberal Party are now using to defend calls for even tighter immigration policies and national border control," according Wind.

Climate - an EU issue

While immigration and border control have been dominating election debates for nearly two decades, voters are increasingly concerned about the climate.

As a record 66 percent of Danes turned to up to vote at the European elections last week, and parties with strong climate agendas were clearly rewarded: the Socialist Peoples Party and the Red-Green Alliance both gained seats, while the DPP and the Peoples Movement against EU suffered.

This does not come as a surprise to Stine Bosse, chairwoman of the European Movement in Denmark.

"The issue of climate change has overtaken immigration as the top priority on the electoral agenda. Danes now seem to agree that we cannot solve this on our own and that we need to work with the rest of the EU member states to fight climate change - as fast as possible," she said.

"All mainstream parties are now pushing the climate agenda, as they are realising that voters are actually concerned. This will no doubt serve to create a stronger pressure within the EU system," said Bosse.

And according to professor Wind, climate change is giving new strength to the EU: "Even anti-EU parties and politicians, who have previously been going on about whether to stay or leave, are now discussing how we can actually use the EU. This is also reflected during discussions in our parliament."

Referendum? - but on defence policy

As Danes prepare to vote for their next government, they might bear in mind that Lars Lokke Rasmussen, the leader the Liberal Party and the current government, recently raised the possibility of a referendum on the Danish opt out on the Common Security and Defence Policy (CSDP).

While Frederiksen, leader of the Social Democrats, immediately ruled out a referendum on the issue, the weakening of the DPP, which could tip the balance for both mainstream parties, might make her change her mind.

"She has not been outspoken about a referendum, but the other day she mentioned that 'talks' might be necessary. Any responsible government being in power for the next four years will have to raise this question at some point, taking the current geopolitical situation into account," said Bosse.

"It all depends on the Social Democrats. A referendum will not happen unless the Social Democrats and the Liberal party agree on it, and the Social Democrats do not believe this is the time," according to Wind.

Just say 'no'

Denmark has already held two recent referendums, of course.

In 2000 Danes voted 'no' to the single currency, and in 2015 they voted 'no' to modifying their home & justice affairs opt-out.