Rep. Paul Ryan, chairman of the House Ways and Means Committee, wants to hold the vote on Trade Promotion Authority this week before members head back to their districts. (Molly Riley/AP)

House Republicans and President Obama joined forces for the final push for votes to keep the White House’s trade agenda alive, unveiling the latest version of fast-track authority overnight Tuesday and setting up a possible Friday vote on the measure.

GOP leaders hoped to allay Democratic concerns about a minor Medicare provision in the sweeping legislation, but the issue remained unsettled after two closed-door party caucuses Wednesday morning.

Union officials, in a pair of strongly worded letters, accused their regular ally — Obama — of having “marginalized” labor views and asked Democrats to vote against a program designed to help laid-off workers in a manner that would kill the entire trade legislation.

Complicating the issues even further is the parliamentary morass House Speaker John A. Boehner (R-Ohio) has charted for the next few days, giving opponents multiple avenues for defeating Obama.

If Boehner and Obama’s advisers are not confident in the outcome, the vote could be pushed into next week, but leading Republicans, particularly Rep. Paul Ryan (Wis.), have been pushing to hold the vote before allowing lawmakers to leave for the weekend — and another round of hometown opponents pushing House members to vote no.

Rep. Gerald E. Connolly (D-Va.), who is supporting the trade proposals and accompanied Obama to Europe last weekend, said the administration has a ­“quiet confidence” that it can win the vote with a strange-
bedfellows coalition consisting of roughly 195 Republicans and
25 to 30 Democrats. “They know it’s going to be close, but they see a path to passage,” Connolly said after the trip, which included two long discussions with Obama and a marathon dinner with his chief of staff, Denis McDonough.

The Senate, on a large bipartisan vote, approved Trade Promotion Authority last month. It would give the Obama administration a more certain process for finalizing a large 12-nation deal to expedite commerce across the Pacific Ocean. At a meeting of the White House Export Council on Wednesday, U.S. Trade Representative Michael Froman said the other 11 nations would not put their best, final offers on the table until the fast-track issue is resolved in Congress.

“They’re only willing to do that if they feel like we’ve got the political support here to move that forward,” Froman said.

The TPA bill includes new funding for a program designed to help U.S. workers who lose jobs because of international competition, known as Trade Adjustment Assistance, which has broad Democratic support but faces some GOP opposition from Republicans who consider it a form of welfare.

Opponents kicked off their final push to defeat Obama’s top remaining legislative priority by latching onto a relatively small cut in Medicare that was meant to offset increased funding for worker training.

“We are going to vote against anything that will cut Medicare,” Rep. Rosa L. DeLauro (D-Conn.) said at a rally outside the Capitol.

The Democrat-on-Democrat tension has risen to levels not seen in the Obama era. Rep. Brad Sherman (D-Calif.) accused Obama of living “in a cloister” where only “captains of industry” get to air their views to him.

In the five-page missive sent Monday, AFL-CIO President Richard Trumka accused Obama of mischaracterizing labor’s stance on the emerging Trans-Pacific Partnership (TPP). “You have repeatedly isolated and marginalized labor and unions as the only opponents of fast track and TPP,” Trumka wrote, according to a copy obtained by The Washington Post. “I am sure you are aware, however, that the critics of the current TPP encompass a broad, deep, and intellectually impressive swath of public opinion.”

Republicans plan to hold several votes on the legislative package — including separate votes for the trade authority and for the worker assistance. If both of those passed, they would be reattached and then sent to the White House for Obama’s signature.

The expectation is that all the Democrats would vote for the TAA legislation, along with a few dozen Republicans from districts where manufacturing jobs have been hurt by global trade, then a huge bloc of Republicans — along with a couple of dozen Democrats — would support TPA.

But labor officials are urging Democrats to sabotage the piece of the legislative jigsaw puzzle that requires huge Democratic support: TAA’s worker-retraining funds, even though it is for a cause they ideologically support. If they take down that vote, they would torpedo Obama’s entire trade agenda.

Boehner and Ryan, chairman of the Ways and Means Committee, do not want to alter the delicately balanced TPA-TAA package because that would require sending it back to the Senate for another vote and potentially several more weeks of debate there. They worked to avert this issue by advancing a separate piece of legislation that would replace the roughly $900 million cut in Medicare, slated for 2024, with some stricter enforcement of tax laws. He told reporters Wednesday that he would allow that legislation to be voted on first so that Democrats could vote for TAA without fear of retribution from their allies for supporting even a modest cut in the popular entitlement program.

“That solves the problem,” Boehner said. “Now, if people are looking for an excuse to vote no, I guess they can always find an excuse to vote no.”

However, in the closed-door huddle of Democrats, Pelosi characterized that plan as “unacceptable” because it left the language of the Medicare cut in the legislation and said that, no matter if it were never going to be enacted into law, Democrats could not support such a plan, according to a handful of lawmakers in the room.

At the moment, Democratic ­opponents expressed confidence that they would have the numbers to defeat the vote for TAA.

“I’ve been whipping on this, and my sense is that we have mostly no votes,” said Rep. Jan Schakowsky (D-Ill.). “I haven’t personally come across anybody who said they were going to vote for it.”

Mike DeBonis contributed to this report.