WASHINGTON — 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.

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.

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.

“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.