The representative, whose name was not reported, told CNN the House, controlled y Republicans, and the Senate, controlled by Democrats, could agree on a bill that would fund the government for the rest of the year and put off the debt limit crisis with a six-week extension on borrowing over the limit.
CNN described the representative as a "senior Republican."
He suggested his Republican colleagues are being "unfair" to conservative supporters by suggesting they can use the shutdown to effectively repeal the Affordable Care Act when they know they can't deliver on the promise.
Democrats, including President Barack Obama, have said they would only hold budget discussions once the shutdown ends, so it is unclear whether they would be willing to support the congressman's plans.
The development came after the House voted Saturday to pay federal workers retroactively when the shutdown ends.
The vote in the House on retroactive pay was unanimous with the Federal Employee Retroactive Pay Fairness Act passing 407-0, The Hill reported.
Republicans said they would vote on more piecemeal funding bills Saturday even though Democrats insist on a comprehensive spending bill.
House Majority Leader Eric Cantor, R-Va., told reporters Friday the House would vote on funding for border protection and Head Start. The House Friday passed bills that will fund the National Institutes of Health, Federal Emergency Management Agency and allow the reopening of national parks, monuments and museums.
The Republicans, who had refused to authorize a larger budget bill that included funding for healthcare reform, have been passing the agency-specific bills as part of a strategy to shift the blame for the government shutdown to the Democrats, The New York Times said Saturday.
Democrats prefer a full spending measure that raises the debt ceiling because piecemeal bills mainly cover programs popular with Republicans and leave others unfunded.
"The issue here is whether we're going to pick winners and losers by providing temporary funding for governmental services, operations and personnel," said Rep. David Price, D-N.C.
Senate Majority Leader Harry Reid, D-Nev., has warned the Senate will not vote on the smaller bills passed by the House and will negotiate with the GOP only if Republicans agree to end the government shutdown with no strings attached.