Reid and McConnell also have been discussing provisions to give federal agencies flexibility in adjusting to across-the-board spending cuts imposed under legislation that Obama signed in 2011.
Another element of their negotiations would call for House-Senate negotiations on a possible deficit reduction measure to take the place of the across-the-board cuts.
The twin crises began more than three weeks ago, when some lawmakers in the House insisted on seeking the defunding of Obamacare as the price for preventing a partial shutdown of the government.
The White House refused, and the Democratic-controlled Senate rejected legislation to achieve the GOP goal, as well as subsequent legislation that contained scaled-back concessions on the health care overhaul.
The partial shutdown, which began on Oct. 1, swiftly merged with the approaching debt crisis.
According to Treasury Secretary Jacob Lew, unless Congress acts by Oct. 17, the government will lose its ability to borrow, and would be required to meet its obligations relying only on cash on hand and incoming tax receipts.
While it was unclear how long that would suffice, the date stood as a deadline. The closer it approached, the more urgent became the pleas of businesses and bankers in this country as well as officials overseas for the United States to put its finances into order.
Whatever the outcome, the all-out assault on Obamacare that became a tea party rallying cry last summer was long gone, repulsed by the president and his Democratic allies in Congress.
Instead, Republican disapproval ratings have plummeted in public opinion polls in the past two weeks, vindicating warnings from Boehner, McConnell and other party elders that the original strategy of threatening to shut down the government in hopes of wiping out the overhaul was badly flawed.
"We got ourselves in a ditch," McCain said. "And we got to stop digging."
Associated Press writers Donna Cassata, Andrew Taylor, Henry C. Jackson, Julie Pace and Alan Fram contributed to this report.