Speaker of the House John Boehner says he will propose a temporary fix that would extend the debt limit for six weeks.
With seven days to go before the nation reaches that limit, the White House called the short-term proposal an encouraging sign.
"I would hope that the president will look at this as an opportunity and a good-faith effort on our part to move halfway -- halfway to what he's demanded -- in order to have these conversations begin," the Ohio lawmaker said Thursday.
The White House press secretary Jay Carney said if a clean bill is passed, the president "would likely sign" it.
"We'll see what they're able to pass and consider it then," Carney said.
Are Republicans trying to work with the president or is the GOP giving up their fight? Phil Kerpen, president of American Commitment, talks about this and more on CBN Newswatch, Oct. 10.
Should the president accept, the extension would not end the partial government shutdown, but it would allow more time for negotiations on a budget deal.
The plan, proposed by Boehner Thursday morning, has yet to be endorsed by the Republican caucus and some of the details are still up in the air. But if there is enough support within the party, a vote could come quickly.
This latest move from the House came shortly after Treasury Secretary Jack Lew testified before the Senate about consequences the United States will face if it doesn't raise the debt ceiling by Oct. 17.
"If Congress fails to meet its responsibility, it could deeply damage financial markets, the ongoing economic recovery, and the jobs and savings of millions of Americans," Lew warned lawmakers.
He said the government is risking its first default in U.S. debt history.
Still, some Democrats say they are only willing to negotiate on the debt ceiling when Republicans vote to end the shutdown.
Meanwhile, President Barack Obama met with congressional Democrats and Republicans separately Thursday to discuss ways to end the partial government shutdown, which is now in its tenth day.