Lew said that members of Congress "need to open the government up. They can do it today."
But Republicans blamed Obama for the impasse.
"What he needs to do is to roll up his sleeves," Sen. John Cornyn, R-Texas, said Sunday on CBS.
Cornyn said the shutdown, which has left hundreds of thousands of government workers at home, has happened 17 times since 1976 and is not a crisis. But he said it cannot end without Obama sitting down with congressional Republicans.
"We're not going to resolve this without the president engaging," he said. "So far, he's been AWOL," he added.
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."