Despite strong bipartisan support in Congress to undo a recently passed cut to military pensions, a debate between the House and Senate over whether to offset the cost of restoring the benefits has muddied its path for progress.

The Republican-led House on Tuesday easily passed a measure to pay for the cost of canceling the pension cuts by extending for an additional year a 10-year "sequestration" cut to Medicare reimbursements to doctors and hospitals.

But Senate Democrats are pressing for a "clean" bill that would undo the cuts without paying for them, and appear to have little political appetite to accept the House's position.

"The thousands of veterans in Louisiana are all wondering why we are debating an offset," said Sen. Mary Landrieu, D-La., a sponsor of the Senate bill. "Whatever was owed they already paid.

"It was promise we made, it's a promise we should keep," she added.

Proposals in both chambers would undo cuts to cost-of-living pension increases for military retirees under the age of 62. The cuts were included in December's budget agreement brokered by Rep. Paul Ryan, R-Wis., and Sen. Patty Murray, D-Wash., who head their chamber's budget committees.

The cuts caught many lawmakers from both parties and chambers off guard, and restoring them has been a bipartisan priority.

The Senate bill cleared a procedural hurdle Monday by a vote of 94-0. But Sen. Kelly Ayotte, R-N.H., has proposed an amendment to offset the pension cuts by preventing illegal immigrants from claiming a child tax credit.

Senate Majority Leader Harry Reid, D-Nev., says the amendment is "ill-advised" and unfair to children, and hinted he may not bring it up for a vote.

"She's changed it, tried to make it better, but it's not better," Reid told reporters Tuesday. "If there is something with fraud, then go after the tax preparers, but not these children."

But denying Republican amendments could cause a backlash from the minority party and pose problems for the bill's survival.

"Our view is that it ought to be fixed by paying for it and not adding to the deficit, and the alternative we expect to be able to offer and to get a vote on is led by Sen. Ayotte," said Senate Minority Leader Mitch McConnell, R-Ky.

Sen. Dean Heller, R-Nev. on Wednesday also filed an amendment that would offset the cost of undoing the pension cuts.

Final passage on the Senate bill could come as early as Wednesday.

Sen. Mark Pryor, D-Ark., the bill's main sponsor, said the House's plan to implement Medicare cuts to offset the cost for restoring the veterans' benefits would put an undue burden on seniors. But he stopped short of saying he wouldn't support the measure if it came to the floor for a vote.

Sen. Mark Begich, D-Alaska, another sponsor of the Senate version, says the upper chamber -- for the sake of veterans -- should pass the measure immediately and worry about paying for it later.

"Let us do what we do in regard to the budget and appropriations and we'll deal with [the offset] issue down the road," he said. "But let's not have these [veterans] hanging out there with uncertainty in regards of their benefit that they have earned and fought for this country."