Looking to break the “fiscal cliff” gridlock, House Democrats are attempting to use a “discharge petition,” a little-known procedural move, to get around Speaker John A. Boehner and force a House vote on extending the Bush-era tax cuts for everyone except individuals making $200,000 and more and couples making $250,000.

Such a vote would give Republicans who want to abandon the speaker and the GOP “no-higher-tax-rates” position a chance to buck party leaders and cut their own deal with Democrats to avert the “fiscal cliff” — the automatic spending cuts and tax increases that will take effect in January.

No Republicans have signed or said they would sign the petition, but Democrats are optimistic they can find more than two dozen GOP lawmakers needed to send the stalled bill to the floor for a vote — especially if negotiations between the president and Mr. Boehner remain stalled.

“The discharge petition, while we do want to get it to the [House] floor, it can also be a catalyst to move people,” said Rep. Timothy J. Walz, a Minnesota Democrat who filed the petition on behalf of his party. “It sends a signal to [House Speaker John A.] Boehner that, if he gets a few of his members on it, it maybe frees him up for a little bit of negotiating.”

The petition calls for House action on a bill that passed the Democrat-controlled Senate this year that would keep George W. Bush-era tax cuts in place for individuals earning up to $200,000 and couples making up to $250,000. For those earning more — about 2 percent of taxpayers — the tax breaks would expire as scheduled in January.

Republicans have expressed little interest in the measure and say they want the tax breaks extended to all income levels.

Democrats were inspired in part to try the unconventional approach after Rep. Tom Cole, Oklahoma Republican, last month said he would be willing to let the tax cuts expire for the nation’s most wealthy as a way to help avoid the looming “fiscal cliff.”

Rep. Walter B. Jones, a moderate Republican from North Carolina, also said he would consider signing the petition.

Discharge petitions require a simple majority of the House’s 435 members, or 218 votes, to advance a stalled bill toward a full floor vote. If all of the chamber’s 191 Democrats sign Mr. Walz’s petition, the support of 27 Republicans would be needed to get the bill to the House floor.

If the required signatures are secured this week, the earliest likely day that the bill could be brought to the floor under House rules is Dec. 24.

Thirteen Democrats haven’t signed the petition, which was filed Tuesday. Mr. Walz said he expects most, if not all, Democrats to give their support. He noted that many who haven’t done so were out of town last week and said no Democrat has refused to sign the petition.

Getting “the 191 Democrats on it, that’s great, that’s nice, that shows unity. But the real story is when you get the first Republican,” Mr. Walz said. “There are some [Republican] members that might simply see this as a way to kind of change the debate.”

Mr. Walz said the petition isn’t meant to shame or coerce rank-and-file House Republicans to accept a tax deal against their party leaders’ wishes. Rather, he said, it offers a generally nonconfrontational path for Republicans who quietly agree with the Democratic measure.

“I’m very cognizant that this is very difficult vote for a lot of colleagues, and I’m not gleeful in twisting that in anymore,” he said. “I want to reach a compromise that works for everybody.”

But Republicans have reason to be leery of crossing the House speaker. Last week, Mr. Boehner took several Republicans — including Mr. Jones — off committees when their voting records weren’t considered in line with the party.

Boehner spokesman Michael Steel said there is no way enough House Republicans will defy the speaker and sign the petition.

“Democrats like to talk about a lot about arithmetic these days, but clearly the arithmetic with the discharge petition doesn’t work,” Mr. Steel said.

“The only obstacle standing in the way of middle-income tax relief are the Republicans’ unwillingness to ask the top 2 percent to pay their fair share,” said House Minority Leader Nancy Pelosi, California Democrat. “This is a moment of truth. The clock is ticking.”

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