House Republican leaders, faced with the task of writing a budget that would eliminate deficits within 10 years, are drawing opposition from party members to the idea of revamping Medicare for more Americans than previously suggested.

House Budget Committee Chairman Paul Ryan (R., Wis.) has in recent weeks floated the idea of rolling back the GOP promise that people 55 and older would be exempt from his signature plan to offer future retirees a subsidy to buy private health insurance as an alternative to traditional Medicare.

House GOP centrists are balking at the idea, and in a meeting with them Tuesday, Mr. Ryan said he would leave it to his party colleagues to decide whether to raise the age cutoff to 56—or even higher—in the budget for the fiscal year that starts Oct. 1 that he is unveiling next week, according to a GOP aide familiar with the meeting.

Rep. Charlie Dent (R., Pa.), one of the centrists who was worried about the proposal, said after the meeting he believed the proposal would be dropped. “There’s a fairly good shot that the budget will maintain no [Medicare] changes for anybody 55 and up,” said Mr. Dent. “That was a concern that I think many members had.”

Mr. Ryan’s staff said the House budget’s Medicare provisions remain in flux, and no final decisions have been made. But the intense backroom debate over Medicare rules underscores the challenge facing Mr. Ryan and his party as they try to make good on their promise to House conservatives to balance the budget within a decade.