Republicans have been demanding cuts in government programs, including Obama's 2010 health care law, and a bigger effort to cut long-term federal deficits as their price for reopening government and extending the debt limit.
Obama has repeatedly noted recent improvement in the deficit figures. After four years of trillion-dollar deficits, the 2013 shortfall is expected to register below $700 billion.
Rep. Vern Buchanan, R-Fla., said the plan was for the House to approve the legislation Boehner described on Friday.
"It gets us down the road a little bit so they can continue to talk," said Rep. Tim Griffin, R-Ark. Rep. Robert Pittinger, R-N.C., said the six-week extension would provide "an opportunity to bring the parties together."
Some conservatives still expressed reservations. "I'm not very enthusiastic about that," Rep. Steve King, R-Iowa, said of Boehner's plan.
Under Boehner's offer, the House would also appoint negotiators to bargain with the Democratic-led Senate over a budget compromise. Those talks have been on hold for months, and the two chambers have deep differences over taxes and cuts in benefit programs.
Earlier Thursday, Treasury Secretary Jacob Lew warned the Senate Finance Committee that failure to renew the government's ability to borrow money "could be deeply damaging" to financial markets and threaten Americans' jobs and savings. It would also leave the government unsure of when it could make payments ranging from food aid to Medicare reimbursements to doctors, he said.
"The United States should not be put in a position of making such perilous choices for our economy and our citizens," the secretary said. "There is no way of knowing the irrevocable damage such an approach would have on our economy and financial markets."
The game of Washington chicken over increasing the debt limit — required so Treasury can borrow more money to pay the government's bills in full and on time — already has sent the stock market south, spiked the interest rate for one-month Treasury bills and prompted Fidelity Investments, the nation's largest manager of money market mutual funds, to sell federal debt that comes due around the time the nation could hit its borrowing limit.