“Obama’s ability to sway the debate about free trade has been hampered by those in his own party feeling the heat from constituents who are up in arms about weak employment and wage growth,” Dr. Prasad said.

White House officials said Mr. Obama would not hesitate to make a strong case for the Trans-Pacific Partnership on the campaign trail. But Josh Earnest, the press secretary, said the president’s remarks on Tuesday in Charlotte would probably focus more on areas where he and Mrs. Clinton agreed.

Then, in Europe, Mr. Obama may find a more receptive audience, given the deep misgivings over the British vote.

“His timing is pretty good,” said David Axelrod, a former senior adviser to Mr. Obama. “If he arrived the week before, the party would have been rip-roaring. Now everybody is sitting there with an ice pack on their forehead. They might be ready to listen to some sensible advice.”