Republican leaders say such voices represent a fringe in the party. After all, trade promotion has been embraced by longstanding conservative voices like the anti-tax gadfly Grover Norquist and the editorial page of The Wall Street Journal. But they cannot afford to lose much of that fringe, given the firm Democratic position.

With organized labor, environmental groups and liberal advocacy organizations like MoveOn.org opposed to the bill, Democratic leaders say they are in no position to deliver votes for Republican leaders, even if their own standard-bearer in the White House is adamantly behind the measure as well. The A.F.L.-C.I.O. began running attack ads against one Democrat, Representative Ami Bera of California, who came out for trade promotion authority and who has twice used union muscle to eke out narrow victories in his swing district. Less pointed pressure ads are going up in 17 other Democratic districts.

“It doesn’t make any sense for our members to make a career-changing vote” to let Republican members off the hook, a senior Democratic leadership aide said.

That has Republicans — and the president — in a bind. Representative Jim Jordan, Republican of Ohio and leader of the conservative House Freedom Caucus, said that as written, the trade promotion legislation lets Congress “turn off” fast-track approval if a trade deal comes back having failed to meet the standards demanded in the bill.

But in reality, he said, the process for shutting down fast-track is so byzantine, it is nearly impossible to do. Such a resolution would first have to go through the House Ways and Means Committee and Senate Finance Committee, which tend to be pro-trade, before it could be considered by the House and Senate. Mr. Jordan and a handful of like-minded colleagues plan to amend the trade bill to make it easier for Congress to halt fast-track.

“I understand economics. I understand comparative advantage. I understand livestock producers in Ohio who want to sell pork in Japan,” said an exasperated Mr. Jordan, whose vote could influence a dozen or more conservatives. “But I’ve got a real problem here.”

Mr. Ryan, the manager of the bill, said there was no way he could allow such an amendment. That would make the House’s trade bill different from the Senate’s, prompt a negotiating conference between the chambers and force both to have to vote all over again.