Noting the House had passed more than a dozen bills to fund "things we can all agree on," Rep. Buck McKeon, R-Calif., urged Obama and Senate Democrats to "back these bills immediately."
"Then the president should work with us on plans to reopen the entire government and make sure we do not default on our debts," said McKeon, delivering the GOP's weekly media address.
McKeon voiced optimism about a meeting he and other House Republicans had with Obama at the White House Thursday.
"For all the focus on disagreements in Washington, we've actually found some common ground this week," he said, pointing to legislation Congress passed and the president signed guaranteeing death benefits to the families of military service members who die in action.
"We shouldn't stop there," McKeon said.
He made no mention of defunding the Affordable Care Act, popularly known as Obamacare. House Republicans had linked defunding or changes in the healthcare law to negotiations on raising the debt limit.
The negotiations, McKeon said, were about "making sure there's fairness for everyone."
"And the longer we go on settling for maybe-next-time, for this notion that putting things off until after the next election is okay, the harder this will get," McKeon said in a transcript provided by ABC News Radio. "It's a challenge, but we can do this. Let's get back to work, together."