HARRISBURG — With Pennsylvania’s budget stalemate in its fifth month, Democratic Gov. Tom Wolf and leaders of the Legislature’s huge Republican majorities say the momentum of closed-door talks has picked up and there is cause for optimism.

One key development is the identification of a source of money to help prop up the budget: diverting slot-machine gambling revenue into the state treasury.

However, differences remain between Wolf and Republican lawmakers, and Republicans may have to confront their own internal differences to pave the way to a broader agreement.

Wolf’s press secretary, Jeff Sheridan, said Saturday there are serious negotiations and a light at the end of the tunnel. That followed an upbeat note Friday by House Majority Leader Dave Reed, R-Indiana, to rank-and-file Republicans saying progress in talks could lead to the passage of a budget by Thanksgiving.

“We’re not there yet, but for the first time in a while, we seem to have a direction that we’re heading in together,” Senate Majority Leader Jake Corman, R-Centre, said.

Wolf has sought a multibillion-dollar tax increase to correct a long-term deficit and narrow a funding disparity between rich and poor school districts that’s considered among the nation’s widest. Meanwhile, House Republican negotiators want Wolf to agree to privatize the state-controlled wine and liquor system and Senate Republicans want Wolf to agree to replace the traditional pension benefit for future school and state employees with a 401(k)-style plan.

Republicans have thus far united against a tax increase to support state government operations or public schools. As of Friday, there was no agreement on how to meet Wolf’s funding demands, Wolf administration officials said, and Republicans have not agreed to one of Wolf’s priorities: increasing taxes on Marcellus Shale natural gas production.

House Speaker Mike Turzai, R-Allegheny, has maintained his opposition to it, although Corman and Reed had not ruled it out as a concession to Wolf.

One new idea under discussion could provide a cornerstone of the cash Wolf is seeking: diverting the roughly $600 million in slot-machine gambling receipts that school districts currently pass along to homeowners as property tax reductions. That stream of money would shift to the state treasury, while negotiators are discussing an increase in the state sales tax to offset reductions in local school property taxes.

The new money from a higher state sales tax would replace the diverted slot-machine dollars to schools — and then some.

But many disagreements remain.

For one, negotiators report differences over how much of an increase in aid the state should sent to public schools. Wolf will not sign onto a budget agreement without a historic increase, Sheridan said Saturday.