WASHINGTON — Republican lawmakers said Thursday that they would support legislation allowing the Department of Veterans Affairs to shift money from another program to cover a $2.5 billion budget shortfall that would otherwise threaten medical care for many patients in coming months.

Representative Jeff Miller, Republican of Florida and chairman of the House Veterans Affairs Committee, said at a hearing on Thursday that he would work with other lawmakers to help cover the budget gap.

The money would come from a new program called the “Choice Card,” which allows certain veterans on waiting lists and in rural areas to choose taxpayer-paid care from private doctors outside the department’s health system. The program has been a priority for Republicans, who say using private doctors will help reduce health care waiting lists for veterans and will inject a measure of competition into an inefficient agency.

The shortfall surfaced just months after lawmakers were told that the agency was actually under budget for the fiscal year. But soaring demand for medical care from veterans has exceeded even the most aggressive predictions about how much, and how quickly, the agency would have to grow to put last year’s waiting list scandal behind it.