The Senate’s push to repeal the Affordable Care Act has bogged down in the face of a deep divide among Republicans, and some senators are expressing vastly disparate reservations about what the bill does and does not do.

But there is another contentious element of the plan, found in both the House and Senate versions, that has received scant attention in Washington, mostly because it affects only one state: New York.

That is the so-called Faso-Collins amendment, which would end a funding formula that for 50 years forced New York’s 62 counties to help pay for Medicaid, the public health insurance program largely for low-income people.

The state’s counties currently pick up 13 percent of the total cost of Medicaid, far more than counties in other states pay, if they pay anything at all. The amendment, sponsored by Representatives Christopher C. Collins and John J. Faso, both upstate Republicans, would eliminate the burden for the counties — excluding New York City’s five boroughs.