Senior House Democrats are warning that they could withhold pivotal votes Thursday on the emerging budget agreement if Republican leaders don’t agree to put a three-month extension of unemployment insurance into the package.

“This puts the bill at risk,” Rep. Sander M. Levin said, speaking with reporters on Wednesday afternoon. The Michigan Democrat is the ranking member of the House Ways and Means Committee.

This would be a shift for some Democrats who said late last week they would not make their votes on a budget deal, negotiated by House Budget Chairman Paul D. Ryan, R-Wis., and Senate Budget Chairwoman Patty Murray, D-Wash., contingent upon being allowed to vote to extend emergency aid to out-of-work Americans.

However, GOP leaders at the last minute instructed the Rules Committee to tie a vote on the budget deal to three months of relief for physicians who faced looming Medicare payment cuts — but not to the unemployed.

That relief was not part of the original agreement, Democrats argue, and if lawmakers are going to make an exception for the “doc fix,” why is there no similar exception being made for unemployment insurance?

“This does add a new dynamic that could upset the apple cart,” said House Budget Committee ranking member Chris Van Hollen, D-Md., standing alongside Levin at a last-minute news conference in the House Press Gallery. “It’s going to be a new factor that will have to be discussed in our caucus.”

Van Hollen and Levin have filed an amendment with the Rules Committee that would let the budget deal come to the floor on Thursday with a three-month doc fix alongside a three-month unemployment insurance extension, paid for by cutting direct payments on agriculture subsidies.

Speaker John A. Boehner, R-Ohio, said at his weekly news conference on Wednesday morning that he was open to allowing a vote to extend unemployment insurance if he saw a proposal that was paid for, plus included “other efforts that will help get our economy moving once again.”

“I’ve not seen a plan from the White House that meets those standards,” Boehner said.

It’s unlikely that the Van Hollen-Levin amendment will be accepted by the Rules Committee, but it’s also unclear what that means for Democratic support. Lawmakers on the minority side of the aisle seem to be keeping their powder dry, with few saying outright whether they will vote for or against it.

Democrats also aren’t planning on doing a formal whip count, House Minority Whip Steny H. Hoyer, D-Md., confirmed on Wednesday afternoon, which doesn’t give Republicans a clear sense of how many votes they have to work with in the event hard-line conservatives threaten passage by voting “no.” Many members who fall into that camp are also playing their cards close to their chests.