Speaker of the House John Boehner (R-Ohio), seen here at a Republican Conference meeting on Feb. 4, is pitching linking a debt-limit hike to military benefits. (J. Scott Applewhite/Associated Press)

House Speaker John A. Boehner scrambled to sell a new debt-ceiling solution to his Republican colleagues on Wednesday, encouraging them to demand a restoration of recently cut military benefits in exchange for a one-year extension of the federal government’s borrowing authority.

Though Boehner (R-Ohio) did not formally endorse the idea as his own, he did ask his lieutenants to test it among rank-and-file ­Republicans.

Boehner’s inner circle said he is casting about to find a solution that can pass the House without rupturing the fractious Republican conference, in which disagreements over past debt-limit strategies have caused considerable turmoil. He also wants to avoid a dramatic partisan fight with the White House, which has long resisted GOP attempts to extract major concessions on the debt ceiling.

“Right now, Jesus himself couldn’t be the speaker and get 218 Republicans behind something, so I think Speaker Boehner is trying his best to come up with a plan that can get close to that,” said Rep. Patrick J. Tiberi (R-Ohio), a longtime Boehner ally. “Whatever we move, there will be critics everywhere, but at the end of the day we still have to govern.”

This week, Treasury Secretary Jack Lew said the government will run short of cash to pay its bills by the end of the month unless Congress grants additional credit authority.

The U.S. is the only big country with a debt ceiling. The Post's Karen Tumulty explains why. (The Washington Post)

“Unlike other recent periods when we have had to use extraordinary measures to continue financing the government, this time these measures will give us only a brief span of time,” Lew said Monday in a speech at the Bipartisan Policy Center. “Given these realities, it is imperative that Congress move right away to increase our borrowing authority.”

Outside the House chamber Wednesday night, the plan was gaining momentum, with dozens of GOP members saying they could back it. The benefits for retired military personnel were reduced in last year’s bipartisan budget agreement, which cut $6 billion in payments to veterans over the next 10 years.

“I’d support it in a heartbeat,” said Rep. Doug Lamborn ­(R-Colo.). “We need to figure this thing out, and that’s a way to do it.”

Rep. Virginia Foxx (R-N.C.) said, “A lot of people are supportive right now, and this idea could be a vehicle that wins backing from a lot of Republicans.”

The idea emerged after support for two earlier proposals fizzled. One would have involved another attempt to repeal parts of President Obama’s signature health-care law, while the other would have tried to force Obama’s hand on approval of the Keystone XL oil pipeline.

According to two people pres­ent, Boehner argued that the military-benefits maneuver could force some Democrats to join Republicans and also win support from conservatives, who have voiced frustration with the reductions in payouts for retired military personnel.

Boehner’s approach — moving toward a possible tweak to federal pension funding — signals a departure from past debt-ceiling debates, in which Republicans have frequently demanded sweeping conservative measures in exchange for an extension. It also underscores Boehner’s desire to avoid a partisan standoff with the White House ahead of the midterm elections.

In recent years, debt-ceiling debates have been a chance for House Republicans to try to leverage concessions from the White House and Senate Democrats in the ongoing battle over spending and deficit control. But after October’s government shutdown, many in the House say they have little appetite for another standoff. Some members say they just want to get it behind them. “Look — we owe the money and we’ve got to do something,” said Rep. Peter T. King (R-N.Y.), a longtime Boehner supporter. “It’s time to end the drama and get it over with.”

Others say that, in the end, Republicans will have to pass a “clean” bill, one with no strings attached, so they should just move on it now. “It’s going to end up being clean anyway,” Rep. Justin Amash (R-Mich.) said Tuesday. “I don’t see anything they can put on the table that I would support as some sort of trade-off.”

Even tea party favorites are giving Boehner room to pursue options considered less than optimal by House conservatives.

“There is a pragmatism here,” said Rep. Michele Bachmann (R-Minn.). “You’ve got to know when to hold them and when to fold them. My assessment is that most of us don’t think it’s the time to fight.”

Several Democratic aides, who requested anonymity in order to discuss internal matters, said House Democratic leaders will probably balk if Boehner moves ahead with the plan, since Democrats have insisted that they will not negotiate over the debt ceiling. But they did not rule out the possibility of some Democrats supporting such legislation, amid the clamor in both parties to restore the cuts.

This week, aides to Senate Majority Leader Harry M. Reid (D-Nev.) said the Senate will soon bring forward its own legislation to restore the benefit cuts, taking up a spending bill sponsored by Sen. Bernard Sanders (I-Vt.), the chairman of the Veterans’ Affairs Committee. House Republican aides said Reid’s overlap with Boehner’s potential plan is helpful, especially as Republican leaders look for a debt-limit demand that could win Democratic votes.

But the House GOP’s path is far from settled, in spite of Wednesday’s turn toward a new proposal. On Monday, House Republicans, in informal whip counts taken by the leadership, rejected the two previous options for the debt-limit ­discussions. Those measures had mixed support, and the new ­military-benefits pitch will face similar challenges, with the House’s conservative bloc uneasy with any legislation that would extend the borrowing limit.

One key concern raised late Wednesday by House Republicans: making sure a restoration of benefits is balanced by cuts to other federal programs, in order to not have the measure be cast as a spending increase by watchdog conservative groups that are closely watching Boehner’s playbook. There have also been grumbles about whether a change to the budget deal would violate the carefully crafted terms hashed out in December by Congress’s budget chairmen, Rep. Paul Ryan (R-Wis.) and Sen. Patty Murray (D-Wash.).

House Republicans are looking at a handful of other options, should the military-benefits idea fizzle. Those include the “doc fix,” which would alter the way doctors are reimbursed for Medicare treatments, as well as changes to the federal budget that would reduce fraud and abuse or reduce mandatory spending levels, with the latter championed by the Republican Study Committee, a conservative House caucus.

But if House Republicans cannot find common ground soon on its debt-limit strategy, Boehner’s inner circle acknowledged Wednesday that a clean debt-limit hike — without strings attached — could be in the offing.