House Rules Committee Chairman Pete Sessions, R-Texas, earlier said the House would not vote Tuesday on ending the shutdown or extending the federal debt limit.
A Judiciary Committee session to set rules on the measure, which would have extended the Treasury Department's borrowing authority through Feb. 7, was postponed earlier.
The bill, which followed a Republican conference earlier in the day, also would end subsidies under the Affordable Care Act for elected Washington officials and their staffs.
The federal government runs out of borrowing authority Oct. 17, in little more than a day. The shutdown started Oct. 1.
House Speaker John Boehner, R-Ohio, earlier in the day lamented discord in his caucus.
"There are a lot of opinions about what direction to go. There have been no decisions about what exactly we will do," he said after a closed-door caucus. "But we're going to continue to work with our members on both sides of the aisle to try to make sure that there's no issue of default and to get our government reopened."
Boehner said leaders were talking to members from both parties "to try to find a way to move forward today [Tuesday]."
There was little hope the new House measure would win bipartisan support and it was unclear whether Boehner had the votes to approve the measure to give Senate Republicans more leverage in their discussions with Senate Democrats.
The Washington Post and The Hill said the measure would fund federal agencies through Dec. 15, setting up yet another showdown just before Christmas. It would also give the Treasury Department borrowing authority through Feb. 7.
"The House will vote tonight to reopen the government and avoid default," The Hill quoted Boehner spokesman Michael Steel as saying. "After listening to members at conference this morning, House Republican leaders will bring a plan to the floor which will end the Obamacare subsidies for elected officials and staff in Washington ... and pressure Senate Democrats to accept more sensible dates for the [continuing resolution] and the debt limit."
"We're trying to find a way forward in a bipartisan way that would continue to provide fairness to the American people under Obamacare," Boehner said.
House Majority Leader Eric Cantor, R-Va., said House Republicans wanted Democrats to "sit down and talk to us so we could work out our differences and we've been saying since day one that we believe there should be no special treatment under the law [Obamacare] and there should be fairness for all Americans, those elected and those unelected."
Cantor said he was "glad" Reid and McConnell were talking, but said the House GOP conference also thought it was "very, very important for us to stress in whatever proposal we move forward ... [to] reflect our position on fairness, that no special treatment for anybody under the law."
House Democrats separately assailed House Republicans in general and a group of Republicans that, so far, is adamant about extracting deep concessions on the president's signature healthcare law.
"It is time for reasonable men and women on the Republican side of the aisle in the House of Representatives to step forward and tell the speaker that regardless of the small minority in their party maybe those who are enabling that small minority, it's time to push them aside and to do what is right and just for the American people and for the world," said Rep. Joseph Crowley, D-N.Y., the vice chairman of the House Democratic Caucus.
Rep. Xavier Becerra, D-Calif., who heads the House Democratic Caucus, said while the particulars of the Senate bill weren't known, "we know that they are talking and while they may disagree, what they're willing to do is try to come up with solutions, even though they may disagree."
"[At] least they're demonstrating that they're open to different or new ideas," Becerra said.
When all is said and done, Crowley said, Boehner "has the responsibility to bring to the floor the legislation that will pass, and that will avert what we believe is an irresponsible approach to government."
The White House weighed in on the House plan and quickly rejected it, saying it was written "to appease a small group of Tea Party Republicans who forced the government shutdown in the first place."
Under the Senate plan, a provision to delay or repeal a tax on medical devices was expected to be omitted, Senate aides said. But it was expected to include a year delay of another tax, known as the reinsurance tax, paid by employers.
It would also set up longer-term budget talks, with the goal of reaching an agreement by Dec. 13. A key part of those talks would be to replace a second round of deep automatic spending cuts known as the sequester that take effect Jan. 15.
The Senate GOP caucus was to meet Tuesday to consider the emerging deal, The Washington Post reported.
"Democrats and Republicans in the Senate have been working in a bipartisan, good-faith effort to end the manufactured crises that have already harmed American families and business owners," White House spokeswoman Amy Brundage said in the statement. "With only a couple days remaining until the United States exhausts its borrowing authority, it's time for the House to do the same."