“The endgame is a long game,” Mr. Froman said in a recent interview, “but we’re in that endgame.”

At stake is a colossal trade agreement that would stretch from Peru and Chile to Japan and Vietnam, accounting for 40 percent of the world’s economic activity. It would not just lower tariffs: The pact would require rigorous regulations on labor and environmental standards, as well as the first rules for state-owned enterprises like those run by the governments of Vietnam and Malaysia.

The T.P.P. has emerged as the linchpin of Mr. Obama’s strategic shift to Asia, giving the United States a way to counter the economic inroads made in the region by a rising China. The deal is supposed to be followed by the Trans-Atlantic Trade and Investment Partnership with Europe, though those talks have much farther to go.

“If the United States succeeds in these trade negotiations, and I think we will, Mike would have forged some of the most important institutions that the president will leave as a legacy,” said Thomas E. Donilon, a former national security adviser, whom Mr. Froman served as a deputy before being named trade representative in 2013.

A California-born lawyer who has known Mr. Obama since they were classmates at Harvard Law School, Mr. Froman, 52, exudes a genial charm. But it masks a relentless drive that propelled him from senior posts in the Clinton administration to a career at Citigroup, where he earned millions of dollars before resigning to join the Obama administration.

Caroline Atkinson, his former deputy at the White House, describes his style as a “New York hand in a California glove.”