CNN — Warning that "the clock is ticking," House Democratic Leader Nancy Pelosi on Sunday threatened to bring about a House vote on a bill that would extend the Bush-era tax cuts for families making $250,000, but would allow the tax breaks to expire for those above the threshold.

The Democratic-controlled Senate already passed its version of the bill in July, with strong support from President Barack Obama.

"If Speaker Boehner refuses to schedule this widely-supported bill for a vote, Democrats will introduce a discharge petition to automatically bring to the floor the Senate-passed middle class tax cuts," Pelosi said in a statement.

Under a "discharge petition," a bill can be brought to the floor without going through a committee or without approval of House leadership. The bill would need an absolute majority - 218 votes - to pass.

While there are only 192 Democrats in the House, some Republicans have expressed support for the bill.

The tax cuts are set to expire at the end of the year and are the largest point of contention in the deficit-reduction debate. Boehner and Republicans want to extend the cuts for all Americans, while Democrats want to do so for about 98% of the public, allowing taxes to increase on the top 2% of income earners.

Republican Rep. Tom Cole last week made headlines when he urged his party to pass the tax cut extension for the 98% of Americans now and deal with the rest at a later date. Boehner sharply disagreed with Cole, but a few other Republican members subsequently sided with Cole.

The Oklahoma congressman said he would continue to discuss his suggestion with colleagues, and predicted that if it came up for a vote, it would pass the Republican-led House.

Recent polls show public support for such a bill. According to a CNN/ORC International Poll released last week, 56% of people said taxes should be kept high to help lower-income people, while 36% said taxes should be kept low to create jobs.

With the looming end-of-the-year deadline, Pelosi stressed a sense of urgency in her statement Sunday.

"We must find a bold, balanced and fair agreement to avoid the fiscal cliff. The clock is ticking and stalemates are a luxury we cannot afford."