AUGUSTA — Lawmakers negotiating the state’s next two-year budget huddled behind closed doors with one another and Gov. Janet Mills late last week as they worked to strike a deal on her $8 billion spending package.

Members of both the Democratic majority and the Republican minority said they believed an agreement could be reached well before June 30 – the end of the state’s fiscal year and the deadline for passing a budget to avert a government shutdown.

“We’ve picked all the low-hanging fruit,” said Sen. Jim Hamper, R-Oxford, his party’s lead negotiator on the Appropriations and Financial Affairs Committee. He said some key and controversial parts of the budget, including funding for an expansion of Medicaid, approved by voters at the ballot box in 2016, had been agreed to.

The appropriations committee met throughout the weekend on the budget, but as of Sunday night had not announced a deal.

Lawmakers have settled on the amount they would add to the funds going to general purpose aid for public schools, Hamper said. Those items total about $240 million of the $8 billion package.

There also appears to be agreement that the next two-year budget, which is up 11 percent over the current budget, will not include any new or increased taxes for Mainers, a promise that Mills, a Democrat, made to voters.

Hamper said meetings with Mills on Friday were productive but he declined to offer specifics, noting, “We realize there is still going to be quite a bit of give and take before it’s all said and done.”

Rep. John Martin, D-Eagle Lake, a longtime legislator on the budget panel and former speaker of the House, said there were still minor sticking points in the budget, including a Mills initiative that would boost starting teacher pay to $40,000 a year statewide.

“Of course, there are all kinds of bits and pieces all over the place,” he said. “But I think they are all doable.”

Martin said he believed lawmakers could have a final deal in place within the next few days. He said Republicans were still pushing to lower the budget to something less than the $8 billion sought by Mills and quipped that he had made a $7.9999 billion counteroffer.

While Democrats hold the majority in both the House and the Senate, they need Republican votes to reach the two-thirds threshold in both bodies that would allow the budget to go into effect when the current budget expires. The Maine Constitution requires a balanced two year budget or the government is forced to close until that deal is reached.

Democrats could have pushed through a majority-only budget, but would have had to do that and adjourn the Legislature in April so the budget bill could become law in time for the June 30 deadline. They need Republican votes for an emergency preamble that would allow the budget bill to become law as soon as it is signed by Mills.

Mills presented her budget proposal in February and called the measure “sustainable” even though it boosted spending. She said the state’s tax revenue forecasts show revenues will grow enough in the coming year to sustain the spending package, which includes many initiatives like the Medicaid expansion that were frequently beaten back by her Republican predecessor, Paul LePage.

While negotiators are being tight-lipped on most details, key sticking points include how much more sales and income tax revenue the state should share with cities and towns.  Some lawmakers want revenue sharing, as it is known, restored to a full 5 percent of all the state takes in, while others have offered to bump it to 3 or 3.5 percent, which is an increase over the current 2 percent.

Lawmakers are also at odds over the best way to provide property-tax relief to Maine residents, with several competing proposals still being debated.

The budget-writing committee had set a goal of Memorial Day to reach an agreement, but that date has already passed, and it’s not clear when a budget deal will be reached.