The measure faces a wall of opposition from health care providers, patient advocates and retirees, and has been derided by many Senate Republicans, who are all but certain to reject vast portions of it should it clear the House. But clearing the House is a necessary step to keep alive the Republican promise — seven years in the making — to dismantle President Barack Obama’s signature domestic achievement.

Mr. Upton predicted that the bill was “likely” to pass the House, a tremendous reversal of momentum for a measure that has twice been pulled back from a vote for lack of support.

Their announcement gave a big lift to Mr. Ryan and other Republican leaders as they tried to round up enough votes to push the bill through the House this week.

“We’ve got some momentum,” Mr. Ryan told a Wisconsin radio station on Wednesday morning.

Democrats and health care groups tried to slow that momentum. The liberal health advocacy group Families USA said another $8 billion would do little to improve the “high-risk pools” that could be set up by states to provide coverage to people with pre-existing medical conditions who could not find affordable insurance in the open market.