MIAMI — The federal government opened the door on Thursday to a compromise that could ease Florida’s budget impasse, rooted in a disagreement over Medicaid expansion and the cost of caring for the uninsured.

In a letter to Florida’s top health care official, the Obama administration said that it could authorize $1 billion for the 2015 fiscal year and $600 million for the 2016 fiscal year to reimburse hospitals for treating patients who do not have insurance. The cost would be shared by the state and federal governments. The $1 billion offer is less than half of what the state requested for this year for the Low-Income Pool program, which is set to expire June 30.

Before it could receive the money, the state would have to resubmit its proposal to reflect the lower amounts, and then final federal approval would need to be granted. The administration said it had made the announcement because it recognized that Florida needed guidance to complete its budget.

Payments to hospitals for uncompensated care have caused deep divisions between the Florida House and Senate, which are both controlled by Republicans. The Senate is pushing for a version of private-sector Medicaid expansion. But Gov. Rick Scott, a Republican who announced his support for Medicaid expansion in 2013 despite his opposition to the Affordable Care Act, reversed his position this year and, along with the Florida House, now opposes expansion.