But passage of the House leadership plan will likely need Democratic support after a number of Republican lawmakers expressed skepticism about the cost of the proposal.
Members of both parties have shown support for canceling the 1 percent reduction in cost-of-living increases for non-disabled military retirees of working age that was approved only in December. Earlier on Monday, the U.S. Senate voted 94-0 to advance a similar measure past a procedural hurdle.
"It's the only proposal I think that they felt that enough Republicans and Democrats would vote for," Republican Representative Matt Salmon said after party leaders unveiled the plan at a closed-door meeting on Monday.
House Republicans previously considered various demands surrounding a debt limit extension, including ordering approval of the Canada-to-Texas Keystone XL oil pipeline and repealing some provisions of the Affordable Care Act, Obama's signature healthcare law.
(Read more: Lew warns Congress of February debt ceiling deadline)
But those were dropped because there would not be enough Republican support to overcome the opposition of Democrats, Republican aides said.
Salmon, who is from Arizona, said he believed as many as 150 Republicans and 100 Democrats would vote for a debt limit extension linked to military pension relief, allowing it to pass.
House Democrats want to see how many Republicans intend to vote for the debt limit plan before pledging support, a senior Democratic aide said.
The more modest plan offered by Boehner follows three years of partisan battles over the $17 trillion debt limit and other U.S. fiscal deadlines, in which Republicans sought trillions of dollars in deficit reduction, along with delays and denial of funding to Obamacare.
Representative Mo Brooks said Boehner's plan would leave the United States in a "more precarious" fiscal position.
"Right now, we've got a debt ceiling bill that increases spending, which is diametrically, 180 degrees opposite of what we were battling over just two years ago, where the question was how much in spending cuts we were going to get," said the Alabama Republican.
Many Republican fiscal conservatives view the 2011 budget fight as their most effective use of the debt limit as leverage because it led to a deal to cut U.S. discretionary spending by some $2.1 trillion, including automatic, "sequester" cuts.
(Read more: CNBCExplains: The debt ceiling)
Repealing the pension cuts will cost $6.9 billion over 10 years, the Congressional Budget Office said on Monday. Republicans propose to pay for that by extending the sequester for another year, through 2024, for non-discretionary programs.
But Brooks, Salmon and other Republicans said those savings come too far in the future and may never materialize.