Still, a resolution to the impasse itself is nowhere in sight.
House Speaker John Boehner doesn't see one. Asked Sunday how the standoff ends, he was uncertain: "If I knew, I'd tell you."
The Ohio Republican added President Barack Obama can call him any time to start negotiations to end the shutdown. "He knows what my phone number is," Boehner said on ABC.
Treasury Secretary Jacob Lew said Sunday that Congress should act immediately to reopen the government because the votes are there to pass a temporary budget measure.
"There are no winners here," Lew said on NBC. "Every day the government is shut down does real harm to the American people."
Lew said that members of Congress "need to open the government up. They can do it today."
The federal government was partially shut down Tuesday, the first day of the new budget year, after Republicans and Democrats couldn't agree on a plan to continue funding federal agencies.
House Republicans are demanding significant changes to Obama's signature health care law in exchange for reopening the government, a demand that Democrats say is absurd.
"It was time for us to take a stand," Boehner said.
Since Tuesday, the GOP-led House has passed several bills to reopen selected parts of the government. Democratic leaders are rejecting the piecemeal approach, saying the entire government should be reopened and the 800,000 federal workers on furlough put back to work.
Defense Secretary Chuck Hagel ended the argument for most Pentagon civilian employees, ordering nearly all 350,000 back on the job.
Hagel said he based his decision on a Pentagon interpretation of a law called the Pay Our Military Act, which was passed shortly before the partial government shutdown began. Republican lawmakers had complained in recent days that the Obama administration was slow to bring back those workers even though the law allowed it.
In a written statement released Saturday explaining his action, Hagel said the Justice Department advised that the law does not permit a blanket recall of all Pentagon civilians. But government attorneys concluded that the law does allow the Pentagon to eliminate furloughs for "employees whose responsibilities contribute to the morale, well-being, capabilities and readiness of service members."
Hagel said he has told Pentagon officials, including leaders of the military services, to "identify all employees whose activities fall under these categories." He said civilian workers should stand by for further word this weekend.
In remarks to reporters, Robert Hale, the Pentagon's budget chief, said he did not yet know the exact number of civilians who would be brought back to work but that it would be "90 percent plus." He said there are about 350,000 civilians on furlough.
Hale said he hoped that a "substantial number" could be returned to work on Monday but that an exact timetable was not available.
In a rare Saturday session - and an even rarer showing of bipartisanship - the House voted 407-0 to pass a bill to provide furloughed workers back pay. The Obama administration supports the retroactive pay bill and Senate Majority Leader Harry Reid, D-Nev., said he expects the Democratic-led Senate to pass it.
Even a bill that passed without opposition evoked partisan rhetoric.
"Someone try to explain to the American people today that Republicans decided to shut down government on Oct. 1, and on Oct. 5, they decided to pay all those workers, those 800,000 workers that they told, 'Don't come in to work,'" said Rep. Xavier Becerra, D-Calif. "If it weren't so serious it really would be absurd."
House Majority Leader Eric Cantor, R-Va., noted that many House Democrats supported back pay for federal workers but opposed reopening other selected parts of the government.
The standoff is playing out as an even bigger financial crisis looms. The Treasury Department says the federal government will reach the limit of its authority to borrow money on Oct. 17. If Congress doesn't raise the debt limit, the U.S. will default on its obligations for the first time, triggering what many economists say would be an economic catastrophe.
Associated Press writers Robert Burns and Julie Pace contributed to this report.