Rep. Paul Ryan's "Path to Prosperity" budget plan, which passed the House, would have left the current system in place for anyone now 55 or older. Others would get vouchers once they turn 67 to pay for private insurance.
Congressional candidates who did not have the opportunity to vote for the plan have declined to embrace it, the Times reported Saturday.
Rep. David McKinley of West Virginia, one of the few Republicans who voted against the Ryan plan, has been running against it, putting out a constituent flier that says it would "privatize Medicare."
Ricky Gill, who is running against Rep. Jerry McNerney, D-Calif., said there is a "mature debate" about budget and debt issues. But he would not endorse Ryan's proposals, the Times said.
"It would be very difficult for me to endorse any specific plan without having a seat at the table," Gill said.
More senior Republicans say Republican presidential nominee Mitt Romney will be making the ultimate decision on Medicare policy if he wins the November election.
Sen. Orrin Hatch, R-Utah, ranking Republican on the Finance Committee, said Romney does not have to embrace the entire Ryan budget.
"Mitt Romney will carefully go over every aspect of it," Hatch said.