WASHINGTON — With no serious negotiations in sight, a divided Congress slipped closer to a double-barreled fiscal crisis Thursday as House Republican leaders tried to shift the budget dispute to a fight over raising the government’s borrowing limit.

Trying to round up votes from a reluctant rank-and-file, House Republicans said they would agree to increase the debt limit to avert a mid-October default only if Democrats accepted a list of Republican priorities, including a tax overhaul, a broad rollback of environmental regulations and a one-year delay of the health care law. (The administration delayed some provisions of the law on Thursday — story on Page A2.)

At the same time, Speaker John Boehner signaled he was not ready to abandon a spending fight that could shut down the federal government as soon as Tuesday. Asked whether he would put a stopgap spending bill to a vote free of Republican policy prescriptions, he answered, “I do not see that happening.”

President Barack Obama fired back with a broadside of his own. “No Congress before this one has ever, ever in history been irresponsible enough to threaten default, to threaten an economic shutdown, to suggest America not pay its bills,” he said before a friendly audience in suburban Washington.

The bitter back-and-forth was seen as increasing the possibility of a shutdown or default. The Senate faces a critical vote today to cut off debate on legislation to keep the government open. If Democrats muster 60 votes, Majority Leader Harry Reid will move to strip out House language that guts the health care law and pass a stopgap spending bill that funds the government through Nov. 15, without Republican policy prescriptions.

No one is sure how the House would react.

House Majority Leader Eric Cantor of Virginia encouraged Democrats to come to the table. But in their efforts to unify Republicans, House leaders were only widening the partisan divisions. Behind closed doors Thursday, they laid out their demands for a debt-ceiling increase that include the health law delay, authority to overhaul the tax code, construction of the Keystone XL oil pipeline, offshore oil and gas production and more permitting of energy exploration on federal lands.

Still, many Republican backbenchers balked.