WASHINGTON — The House Freedom Caucus, a group of hard-line conservatives who were instrumental in blocking President Trump’s plan to repeal the Affordable Care Act last month, gave its approval Wednesday to a new, more conservative version, breathing new life into Republican efforts to replace President Barack Obama’s health law.

Senior White House officials, led by Reince Priebus, the chief of staff, have relentlessly pressed Republicans to revive the health care push before Mr. Trump’s hundred-day mark on Saturday, and with conservatives falling into line, the bill has a chance to get through the House, possibly as early as Friday.

It was not clear whether conservative support for the revised legislation would be matched by losses in the center, especially among Republicans representing districts won by Hillary Clinton. But the rest of the House Republican Conference was left with a stark choice: Reject the measure and take the blame previously left at the feet of conservatives for undermining a central goal of the administration, or give it the nod, please voters who want a repeal, and risk taking a potentially fatal hit in the next election for approving a measure expected to leave tens of millions of Americans without insurance.

“Members went home and got an earful,” said Representative Tom Cole, Republican of Oklahoma, saying that voters were wondering, “Why can’t you get something done?” Mr. Cole said he was now “cautiously optimistic” that the measure could pass on the floor.