Passage of a comprehensive bill looked impossible after the Democrats’ loss of a Senate seat in Massachusetts. As an alternative, lawmakers in both parties said, some pieces of the bills already passed by the House and the Senate could be pulled out and packaged together in a measure that would command broad support.

The consensus measure would be less ambitious than the bills approved last year. It would extend insurance coverage to perhaps 12 million to 15 million people — and provide political cover to Democrats, who said they could not simply drop the issue after spending so much time and effort on it.

The pared-back approach would cover fewer than half of those who, according to the Congressional Budget Office, would gain coverage under the House and Senate bills. But it would not put the government on the hook for what critics say is a new entitlement, a change that would appeal to some Republicans.