A key Republican ally in Gov. Terry McAuliffe’s bid to expand health coverage for the poor offered a plan Thursday to a break a months-long deadlock on the issue. But he also split with the governor by questioning McAuliffe’s power to keep government running without a budget or to expand Medicaid on his own.

Sen. Emmett W. Hanger Jr. (R-Augusta) offered a proposal to end the General Assembly’s budget-Medicaid stalemate, which threatens to shut down state government if it is not resolved by July 1, the start of the new fiscal year.

Hanger’s plan asks both sides to give a little, divorcing the Medicaid issue from the budget as opponents want but also remaking a state Medicaid commission so it could no longer block expansion. But the nitty-gritty made the plan somewhat more appealing to the GOP-dominated House, which opposes expansion, than to the evenly divided Senate, which narrowly supports it with the backing of Hanger and two other Republicans.

That the plan also came with a strongly worded statement questioning McAuliffe’s expansive view of his powers stoked hopes in the GOP that Hanger might be wavering. But Hanger said that was not the case.

“I’m not, by any means, backing away from my commitment,” Hanger said.

He said he was surely onto a good compromise because he’d gotten “a little pushback” from House Republicans as well as “my buddy Dick Saslaw,” referring to Senate Majority Leader Richard L. Saslaw (D-Fairfax), a leading advocate for expansion.

House Speaker William J. Howell (R-Stafford) and other House GOP leaders cheered the parts of the plan that appealed to them — splitting Medicaid from the budget, taking a limited view of the governor’s powers — but were silent on the rest. Thursday afternoon, McAuliffe said he was “intrigued” by the plan and eager to learn more, but Saslaw was dismissive of it later in the day.

“That’s just one guy’s idea, and it ain’t a majority opinion,” Saslaw said. “The House may love it, but there’s not a lot of traction for it at this point.”

McAuliffe and the Senate support using Medicaid money to provide private insurance to 400,000 low-income, uninsured Virginians. They say that will help needy residents and boost Virginia’s economy at little cost to the state. Under the federal Affordable Care Act, Washington initially promises to pay 100 percent of the cost, with that share gradually reduced to 90 percent.

Republicans contend that Medicaid needs to be changed more extensively before it is expanded. They also fear that Virginia could get stuck with the $2 billion-a-year cost of expansion if an overextended Washington can’t keep its funding promise.

The dispute has prevented passage of a two-year, $96 billion state budget because the Senate includes it in its spending plan but the House does not.

Hanger’s proposal would pull Medicaid out of the budget, robbing the pro-expansion camp of its chief leverage. But it also calls for changing voting requirements for a state Medicaid commission to tilt control toward expansion supporters and away from opponents, who have the upper hand now.

There is a catch for those in favor of expansion: Hanger’s plan calls for the House only to consider the Medicaid commission bill, not pass it. Hanger said the Senate would maintain some leverage since negotiators would work on the budget and the Medicaid commission bill at the same time.

“If you’re working it concurrently, you have to have an element of faith, and you also still hold the cards,” he said. “Even though they’re decoupled, if the other side is not working with you in good faith, you really don’t have to pass the budget, either.”

Hanger issued the plan along with a strongly worded statement countering McAuliffe’s contention that he could keep the state government operating without a budget. Hanger also rejected the idea that McAuliffe could expand Medicaid through executive action — a route the administration has privately explored.

It “would be totally unacceptable for the Administration to assume executive powers to operate Virginia’s government without an approved budget or to expand Medicaid without the concurrence of the legislature,” Hanger wrote.