Canadian Foreign Affairs Minister Chrystia Freeland (C) speaks prior to her meeting with US Trade Representative Robert Lighthizer in Washington, DC, on August 30, 2018.

Ms. Freeland, who has spent the past several weeks commuting between Canada and Washington, arrived here on Wednesday in a T-shirt that read "Keep Calm and Negotiate Nafta." As has become the norm, she departed on Thursday night with no deal in hand. After two long days of talks with Mr. Lighthizer, Ms. Freeland said that they were working through "tough issues" and maintained that striking a fair deal for Canada, not an end-of-the-month deadline, was the driving force behind the discussions.

"Bob and I joke sometimes that we could switch chairs, we know each other's positions so well," Ms. Freeland said this month in between meetings with Mr. Lighthizer.

As President Trump threatens to ink a deal with Mexico by Sept. 30 and leave Canada behind, the two negotiators with vastly different backgrounds, approaches and priorities are under intense pressure to reach an agreement that can pass political muster on both sides of the border. If they succeed, it will be a result of an unlikely alliance as they seek common ground amid the rockiest relations between the United States and Canada in decades.

For more than a year, Chrystia Freeland , Canada's foreign minister, and Robert E. Lighthizer, the United States trade representative, have been locked in intense negotiations to rewrite the North American Free Trade Agreement.

One is a Rhodes scholar, a globe-trotting former journalist and, perhaps one day, a Canadian prime minister. The other is a veteran Republican lawyer who toiled as a congressional staff aide, represented the steel industry and has been cutting trade deals in Washington since the Reagan administration.

But negotiations, tense from the start, have become increasingly strained, with both sides fighting to win concessions and to protect themselves from the appearance of caving politically.

To the chagrin of Canadians, Mr. Trump has publicly — and at times gleefully — berated their country over its treatment of the United States, particularly its dairy farmers, and rebuked Prime Minister Justin Trudeau as "dishonest" and "weak" after the Group of 7 summit meeting in June. He has slapped steel and aluminum tariffs on Canada, claiming their metal imports threaten the United States' national security.

Mr. Trudeau has openly questioned whether Mr. Trump's promises are to be believed while Ms. Freeland has angered administration officials by giving speeches lamenting the decline of Democratic values in the West — a thinly veiled barb at Mr. Trump — and courting free-trade lawmakers on Capitol Hill.

After being promoted to foreign minister last year, Ms. Freeland gave a high-profile speech in Parliament playing up Canada's prominent role internationally in upholding human rights, multilateralism, democracy and free trade. Ms. Freeland suggested Canada was needed to fill the breach as the United States retreated from its role in world leadership. She echoed that theme again during a June speech in Washington, warning, "If history tells us one thing, it is that no one nation's pre-eminence is eternal."

Ms. Freeland told reporters that she gave a version of her speech to Mr. Lighthizer, though she did not characterize his response. Administration officials, who had already begun to sour on Ms. Freeland for her courtship of free-trade oriented members of Congress, were taken aback by what was seen as a direct attack on the president, according to people familiar with the matter.

Ms. Freeland recently received a standing ovation in Canada for her stance, yet opponents called the speech politically naïve, aimed at scoring points domestically when the country's economy is on the line given the United States is its biggest trading partner. They also questioned the "progressive trade agenda" she presented at the outset of the Nafta negotiations, declaring that Canada would be looking for new chapters on women and indigenous rights as well as climate change.

"That was not a smart move. We'd already seen the president withdraw from the Paris climate change accord and the TPP," said Erin O'Toole, a Conservative foreign affairs critic in Canada, referring to the Trans-Pacific Partnership. "Did we really think he'd bake in carbon pricing into Nafta?"

Discussions between Ms. Freeland and Mr. Lighthizer have been described by those in meetings with them as cordial, yet punctuated with disagreements over Canada's dairy protections, a Nafta dispute mechanism and Canadian intellectual property.

Their negotiating styles differ markedly, those people also say. Mr. Lighthizer, who comes to meetings steeped with a historical understanding of how Canada's trade barriers came into existence, is blunt, delivering statements with a gravelly voice. Ms. Freeland asks probing questions and lays out her country's positions with a friendly Canadian lilt.

Ms. Freeland is also Mr. Lighthizer's polar opposite in public. Chatty and outspoken, Ms. Freeland holds numerous briefings with reporters waiting outside the offices of the United States trade representative each day. One morning this month, she handed out Popsicles to the news media. Mr. Lighthizer usually takes the opposite approach, dashing briskly between the offices and the White House, gaze averted.

"Actually I do enjoy hot weather," he said outside his office, in a rare demonstration of small talk.

Those who have known Mr. Lighthizer for decades say he prefers to do his work behind the scenes.

"I don't think he's seeking press or looking for some larger opportunity — his job is to do what he's doing," said William E. Brock, who was the United States trade representative under President Ronald Reagan.

"I think he probably feels that he can get more done by just working on the task," said Mr. Brock, a former Republican senator of Tennessee.

Ms. Freeland, by contrast, has become a popular public figure in Canada. She is known for bicycling to events and slipping into bathrooms to change out of her cycling gear before taking the podium. Together with her husband, Graham Bowley, a New York Times reporter, she juggles three children with help from a rotation of aunts and is known to have meetings at her house that turn into working dinners.

When the Group of 7 ministers were in Toronto last April, she invited them all, along with Foreign Minister Pavlo Klimkin of Ukraine, to her small Victorian rowhouse for a brunch of eggs and waffles prepared by her children. (The children sat for a time among the ministers at the table, Mr. Klimkin told The Canadian Press.) She jots reminders in ink on her hands and pulls out a notebook during casual meetings to scribble notes.

"She's super comfortable in her own skin and very genuine always," says Lawrence Summers, President Barack Obama's top economic adviser who was a professor of Ms. Freeland's at Harvard and has kept in touch ever since.

He added, "The way she is so diametrically opposite to the cultural characteristics of the current U.S. leadership makes her a very strong voice of the international community at the moment."

Ms. Freeland, 50, grew up in northern Alberta on a canola farm and in a feminist, Ukrainian housing cooperative.

She entered Canadian politics with no experience in brokering international trade deals. The revised version of the Trans-Pacific Partnership that Canada signed onto was based on negotiations that took place under the previous Conservative government.

But Ms. Freeland is credited with salvaging a trade pact with the European Union, which took years to negotiate. In 2016, Ms. Freeland, visibly frustrated, walked out of last-minute talks after a regional parliament in Belgium objected to some of the terms under discussion. Her departure helped galvanize Europe, which relented, and the deal was signed shortly afterward.

Mr. Lighthizer, 70, has a far longer history in cutting trade deals. In the 1970s, he worked on trade and tax issues as the chief of staff to Senator Bob Dole on the Finance Committee and later went on to become the deputy trade representative under President Reagan. There he focused on agricultural and industrial issues and in the execution of bilateral trade deals — an experience that has served him well under Mr. Trump.