WASHINGTON — Ohio’s influential Republican governor, John R. Kasich, said on Monday that he could accept a gradual phaseout of the expansion of Medicaid under the Affordable Care Act, but only if Congress provides states with more money than the House health care bill included and more flexibility to manage the health program for the poor.

Mr. Kasich’s statement could prove significant as Senate Republicans try to find near unanimity on a bill to repeal and replace President Barack Obama’s signature domestic achievement. His position points to a compromise that moderate Senate Republicans could embrace — but that could challenge the chamber’s most conservative members.

“I don’t have a problem with phasing down the enhanced federal payments,” said Mr. Kasich, who is working with several other Republican governors from states that have expanded Medicaid. “But it can’t be done overnight, and it has to be done with the resources and the flexibility that are needed so people don’t get left behind. You just can’t be cutting off coverage for people.”

Senator Mitch McConnell of Kentucky, the majority leader, can afford to lose only two Republican senators as he struggles to get 50 votes for a health care bill whose deadline keeps slipping. A vote he had hoped to get by the end of this month may not come until July.