WASHINGTON — The House Republican leadership signaled Tuesday that Republicans would support an essential increase in the nation’s debt limit in mid-October only if President Obama and Democrats agree to delay putting his health insurance program into full effect — a demand that sets the stage for another economically risky confrontation.

The strategy, which Representative Eric Cantor of Virginia, the No. 2 House Republican, outlined to a private caucus of Republicans, underscored the clout of the most militant conservatives, whose demands to repeal or defund the Affordable Care Act have cleaved the party. While the tactic is fraught with risk for Republicans, some conservative lawmakers and groups objected that it did not go far enough in using the looming fiscal deadlines as leverage with Democrats.

The proposal for dealing with both fiscal fights of the fall — on a continuing resolution to keep financing federal operations after the new fiscal year begins Oct. 1, and a mid-October deadline for increasing the Treasury borrowing limit — seemed to reflect House leaders’ belief that Republicans’ bigger political risk is being blamed by voters for a government shutdown, as they were during the Clinton administration.

Many economists and analysts say the bigger economic risk, however, is a failure to lift the debt ceiling, which would leave the Treasury unable to pay creditors and bills that the government already is obligated for, harm the nation’s credit rating and ultimately could cause the first default. Yet that issue is where House leaders are making their more serious stand against the three-year-old health insurance law they call Obamacare. Speaker John A. Boehner of Ohio had already promised “a whale of a fight” with the administration over the debt limit, and Mr. Cantor shared more specifics of the strategy with his colleagues at the party meeting.