“No wonder President-elect Trump realizes that repeal without replace is the real disaster,” said Senator Chuck Schumer of New York, the Democratic leader. “No wonder he has admonished the Congress not to do plain repeal.”

Republicans now have two powerful reasons to “repeal and replace” together: They hope to protect about 20 million Americans who have gained coverage under the law. And they want a politically acceptable judgment from the Congressional Budget Office on the effects of their alternative.

Mr. Trump’s statement last week that a replacement plan should go hand in hand with repeal efforts had already ignited a sense of urgency among Republicans on Capitol Hill. Over the weekend Mr. Trump said he was close to completing a plan to replace the Affordable Care Act with the goal of “insurance for everybody,” but congressional aides said Tuesday that they had not seen an actual proposal.

Republican congressional leaders are trying to put together a plan that could pass muster with the Trump team and also win approval in the Senate under fast-track procedures that would neutralize the threat of a Democratic filibuster.

House Speaker Paul D. Ryan and Senator Mitch McConnell of Kentucky, the Republican leader, met last week with Mr. Price to hash out alternatives, and they have been in close contact with the relevant committee leaders and staff members to begin hammering out ideas that could come into relief at the end of the month, when Republicans have their annual policy retreat.

Stephen Miller, a former Senate press aide and the incoming senior White House adviser for policy, who has been particularly aggressive in presenting himself as the voice of Mr. Trump on all policy matters, has pushed the notion that a plan should move quickly and in tandem with a replacement measure, rather than in a series of smaller bills, congressional aides said.