After weekend negotiations proved fruitless, senators from both parties said they still thought they could reach agreement in the coming days.
With the most unrealistic demands off the table, the two sides were trying to craft a temporary measure that would allow Washington to step back from the ledge.
Republican Senator Roger Wicker was even hopeful of a breakthrough by late last night.
He said: "I'm hopeful we can have something meaningful by the end of the day.
The Treasury Department says it cannot guarantee that the US government will be able to pay its bills past October 17 if Congress does not raise the $16.7 trillion debt ceiling by then.
It is not clear whether Congress can meet that deadline. Even if Republicans and Democrats in the Senate reach agreement, hard-liners such as Texas Republican Senator Ted Cruz might be able to delay a vote for several days.
The House of Representatives also would need to sign off on the plan. Republican leaders there face strong pressure from a vocal conservative flank that is deeply reluctant to make concessions to President Barack Obama and his Democrats.
Many say they will not back any deal that does not undercut Mr Obama's 2010 healthcare reform law, the Affordable Care Act - a non-starter for Democrats.
Republican Representative Matt Salmon of Arizona said: "We've got to draw some lines in the sand now. This, to me, is an epic battle over Washington versus America, and I hope America wins."
Analysts expect that any deal is likely to come down to the wire.
Though the Treasury will likely have enough cash on hand to meet its obligations for a week or so, it might be forced to pay a higher interest rate on the bonds it is due to issue on Thursday.
Banks and money market funds are already shunning some government bonds that are normally used for short-term loans.
In China, the largest foreign holder of US debt, the state news agency Xinhua said it was time for a "de-Americanised world".
US stocks fell at the opening of markets yesterday. The S&P 500 Index was down 0.53% and the Dow Jones Industrial Average was down 0.48% by mid-morning.
The ongoing government shutdown is beginning to weigh on the economy as well.
The hundreds of thousands of federal employees who have been temporarily thrown out of work are likely to get back-pay when the standoff is resolved. However, they aren't getting paid now, forcing many to cut personal spending and cancel holiday travel plans.
Any agreement in the following days would not resolve disagreements over long-term spending and the Affordable Care Act that led to the standoff in the first place.
Despite the objections of rank-and-file conservatives like Salmon, many Republicans are eager to move the discussion away from Obamacare and toward possible spending cuts.
Republican Senator Bob Corker said: "All of us now are talking about spending, which is where we should have been in the first place."
Republicans have floated plans that would push back any possible default for several weeks and keep the government open for several months, but Democrats say that would simply set up another market-rattling confrontation which could spook consumers and further weigh on the economic recovery.
Democratic Senator Mark Begich said: "If we just extend this to January, we'll be right back in the middle of this."