But Ms. DeLauro and other Democratic lawmakers argue that the president is asking for carte blanche to secretly negotiate a trade deal that would cost American jobs, weaken food safety and financial regulations and undermine environmental and labor standards.

The dispute pits Mr. Obama against his own party over what the president has identified as one of the few patches of common ground between him and Republicans that could yield an accomplishment during his final two years in office. Its outcome could determine the fate of the Trans-Pacific Partnership, a complicated 12-nation agreement that would encompass 40 percent of the world economy and is a crucial element of Mr. Obama’s pivot to Asia. It is also animating an intraparty policy fight among Democrats that could shape the 2016 presidential campaign.

Mr. Obama has dispatched top officials, led by his chief envoy, Michael B. Froman, to Congress to quietly make the case for the agreement and build support for winning the trade authority.

“The president has directed us to be responsive to the very important questions that are being asked by members of Congress,” Thomas E. Perez, the labor secretary, said in an interview. “What’s going on now, I think, is part of the healthy give-and-take that should take place when we’re contemplating an important piece of legislation.”

At the same time, Mr. Obama’s advisers see a ripe opportunity to win approval of his trade agenda given that Congress is fully under the control of Republicans who back trade liberalization.