The likelihood of a government shutdown appeared to increase Tuesday after Speaker John BoehnerJohn Andrew BoehnerRift widens between business groups and House GOP Juan Williams: Pelosi shows her power Debt ceiling games endanger US fiscal credibility — again MORE's (R-Ohio) office said a meeting between congressional leaders at the White House did not resolve the standoff over funding the government for the rest of the year. 

A few hours later, Majority Leader Eric CantorEric Ivan CantorBottom line Virginia GOP candidates for governor gear up for convention Cantor: 'Level of craziness' in Washington has increased 'on both sides' MORE (R-Va.) also cast doubt on a deal, and said the GOP was unlikely to change its rules requiring that legislation be made public three days before a vote in order to ease passage of another government funding measure. 


"The question was, if we got to a long-term resolution we could support, would we waive that, and I'm saying I don't even think that's a likelihood and that there would be some need for a bridge to get there," Cantor said. 

According to a readout from the Speaker's office, BoehnerJohn Andrew BoehnerRift widens between business groups and House GOP Juan Williams: Pelosi shows her power Debt ceiling games endanger US fiscal credibility — again MORE told President Obama that the House "will not be put in a box" in the talks over spending cuts despite the looming government shutdown, which would begin after April 8 without at least a new stopgap measure.

Boehner told Obama that his conference will not be "forced to choose between two options that are bad for the country (accepting a bad deal that fails to make real spending cuts, or accepting a government shutdown due to Senate inaction)," the readout said.

"That this is why House Republicans — in lieu of an agreement in which the White House and Senate agree to real spending cuts — are rallying behind a potential third option: a CR that funds our troops through September while cutting an additional $12 billion in spending and keeps the government running for another week," the readout said. 

Cantor said the resolution shows House Republicans are serious about avoiding a shutdown, but the White House rejected the offer, and House Minority Whip Steny Hoyer (D-Md.), who had supported two earlier stopgap measures, said he would oppose this one.

He said the new proposal is "inconsistent" with Cantor's claims that the GOP would fight hard for a long-term compromise rather than fall back on another temporary fix.

"It is an extraordinarily inefficient, ineffective and costly way of doing business, funding the largest enterprise in the world on a weekly basis," Hoyer said. "I will oppose that."

Boehner told his conference about the new stopgap legislation — which contains $12 billion in spending cuts — during a Monday night meeting. The measure would keep the entire government funded for another week, buying time for a deal on a measure to fund the government through the fiscal year. But it would fund the Pentagon through September, meeting calls for the Defense Department to receive longer-term funding.

The measure is difficult for Democrats to support because it would impose $12 billion in cuts in one week and because it includes controversial policy riders, including one that would bar the use of federal and local government funding for abortion services in Washington, D.C.

Cantor said leaders wouldn’t call up the bill for a vote unless they had enough GOP votes to pass it. “We would certainly be confident if the bill were brought to the floor — we have Republican votes to pass it," he said.

A handful of House Republican freshmen on Tuesday said that they would support Boehner's proposal.

"I will support the Speaker on his initiative. ... I think that we absolutely need to fund defense through the rest of this fiscal year," Rep. Rick CrawfordRick CrawfordArkansas legislature splits Little Rock in move that guarantees GOP seats Gas shortages spread to more states Overnight Defense: Administration says 'low to moderate confidence' Russia behind Afghanistan troop bounties | 'Low to medium risk' of Russia invading Ukraine in next few weeks | Intelligence leaders face sharp questions during House worldwide threats he MORE (R-Ark.) said at a press conference on Tuesday morning.

The nine other Republican freshmen surrounding him nodded their heads in the affirmative when asked if they would support the funding bill, even though it includes a short-term funding resolution. Fifty-four House Republicans voted against the last short-term funding measure put forward by GOP leaders.

—Russell Berman contributed.

This story was posted at 11:49 a.m. and updated at 1:49 p.m.