WASHINGTON—House leaders introduced legislation that would pave the way for Democrats to raise the debt ceiling in the Senate without Republican support, as lawmakers closed in on a procedural agreement to resolve the partisan clash over lifting the government’s borrowing limit.

The bill, which also prevents automatic spending cuts that would kick in next year, is the product of weeks of talks between Senate Majority Leader Chuck Schumer (D., N.Y.) and Senate Minority Leader Mitch McConnell (R., Ky.) over the debt limit. Republicans have pushed for Democrats to raise the debt limit without any GOP support, while Democrats have called on Republicans to cooperate on the issue.