The offer, made late Thursday, would allow the House to vote as early as Friday on a six-week extension of the debt ceiling and set up immediate negotiations to end the government shutdown that began Oct. 1, The Hill reported.
The GOP leadership aide said the offer calls for two sets of high-level talks -- one to reopen the government and one on a broader budget deal that would fund the government through 2014 and raise the debt ceiling.
"We are waiting to hear back from The White House," said Rep. Fred Upton, R-Mich., chairman of the House Energy and Commerce Committee who was among the House GOP contingent that met with Obama Thursday. "My sense is we are getting close to getting it done."
The development came as Obama met Friday with Senate Republicans, who were cool to House Speaker John Boehner's proposal Thursday that would raise the debt ceiling for six weeks but keep the government shuttered.
House Republicans said do not expect a response from the White House until after Obama's meeting with Republican senators.
Senate Republicans are floating a plan that would deal with the debt ceiling and immediately reopen the government, The Hill said.
The Treasury Department said the United States will hit its debt ceiling Oct. 17 and will not be able to pay its bills.
House Appropriations Committee Chairman Hal Rogers, R-Ky., said concessions from the White House must be included in any plan to reopen the government, adding that he opposes any plan that includes a yearlong or six-month continuing resolution -- a bill that would temporarily fund the federal government -- that some Senate Republicans have proposed.
"The staff on both sides are trying to isolate the conditions of the CR," Rogers said. "A year or even a six-month CR would be disastrous [because] it is a punt to the executive branch not to exercise judgment as to where the money is spent."
He said it hasn't been decided whether the funding bill would be packaged with the debt limit in one vote.
Rogers indicated his caucus has been influenced by a new Wall Street Journal poll showing support for the GOP has fallen to 24 percent, the lowest since 1989. Other national polls found Republicans are bearing the brunt of public outrage over the budget impasse.
Conservative Rep. John Fleming, R-La., warned negotiators must get something in exchange for reopening the government.
"There aren't the votes for a clean CR, at least among Republicans," he told reporters.
Obama said Tuesday Republicans "don't get to demand ransom for doing their jobs."
He said the White House he is willing to negotiate any budget item -- including healthcare -- but not under the threat of a shutdown or debt default.
"We can't make extortion ... part of our democracy. ... As reckless as a government shutdown is, an economic shutdown caused by America defaulting would be infinitely worse," the president said.
The president dared the House leadership to put a no-strings-attached continuing resolution to a vote, saying he believed there were enough Democrats and Republicans votes to pass the bill, contrary to what the House leadership has said. Several media tallies indicated there were enough votes from both parties to carry a clean CR.