AUGUSTA — Legislative leaders continued negotiations over the weekend and Monday on a budget compromise capable of passing both the House and the Senate in order to avoid a government shutdown.

Gov. Paul LePage, meanwhile, said he would introduce a temporary, monthlong budget to keep state government offices open next month if lawmakers are unable to finalize an agreement before then. But questions remained Monday about whether the Maine Constitution and state laws even allow the type of stopgap budgeting often seen at the federal level.

Early Saturday, members of the Legislature’s Appropriations and Financial Affairs Committee voted out two competing budget proposals. The majority proposal supported by nine Democrats and two Republicans would increase funding for education, expand the Homestead property tax exemption and allow the sales tax to drop back down to 5 percent, as planned. But the proposal did not contain any of the income tax cuts or welfare reforms sought by LePage and the committee’s four House Republican members.

But even committee members, who had spent nearly 150 days trying to craft a bipartisan budget, acknowledged that neither budget would receive the two-thirds support needed from lawmakers in both legislative chambers to pass. As a result, Republican and Democratic leaders continued working on the two primary sticking points: income tax cuts and reforms to the General Assistance welfare program.

House Minority Leader Kenneth Fredette, R-Newport, said the two sides had robust conversations and made some progress, although he indicated that Democrats have yet to move enough to pick up the support of his 68-member caucus.

“We will continue to talk tonight and tomorrow,” Fredette said during one of his numerous trips between the House floor for votes and his office or the offices of other legislative leaders.

House Speaker Mark Eves, D-North Berwick, described the meetings as “substantive discussions.”

“We are committed to finishing the budget now – not kicking the can down the road,” Eves said in a statement later Monday. “We have a bipartisan agreement. The only thing standing in the way of a balanced budget is the House Republicans’ insistence on a tax cut for the wealthy. Leaders continue to meet. Conversations and negotiations are ongoing.”

The window for lawmakers to strike an agreement is fast closing. Leaders from both parties have said both chambers need to pass a budget by June 17 or 18 in order to give the Legislature enough time to respond to an expected veto from LePage. Staff in the Office of the Revisor of Statutes is currently finalizing the language of the budget-writing committee’s two reports, a process that is expected to take a week or longer.

Eves’ allusion to “kicking the can down the road” was a response to comments from LePage that he would introduce a 30-day “continuing resolution” so lawmakers can work on a budget after June 30, according to The Associated Press.

But in June 2013, Attorney General Janet Mills advised legislative leaders that “continuing resolutions” are not an option under Maine’s Constitution and laws, which require a balanced budget. While the LePage administration distributed a June 2013 letter from its own legal counsel suggesting such short-term fixes are possible, Mills’ office stood by the Democratic attorney general’s interpretation.

“Budgeting month to month, even with the two-thirds majority that would be required, when income is erratic and when new contracts and debt payments are due, would call into question the state’s ability to enact a budget that is balanced as required by the Maine Constitution,” Mills spokesman Tim Feeley wrote in an email. “The very suggestion might also raise red flags with the bond market and put our credit rating at risk.”