AUGUSTA, Maine — Maine lawmakers said late Monday they had reached a tentative agreement that could end the state’s government shutdown.

The state Legislature failed to enact a budget Friday, causing the shutdown. Democratic lawmakers and Republican Gov. Paul LePage had sparred over issues including education funding, taxes on high earners and a proposed increase to a lodging tax.

The new proposal would nix an increase to the lodging tax, which LePage has said is critical for him to sign it. Democratic lawmakers said Monday they were willing to make that change in exchange for concessions on Head Start and behavioral health programs.

A spokeswoman for Democratic House Speaker Sara Gideon said lawmakers would vote late Monday or early Tuesday on the new spending plan that repeals the lodging tax increase.

LePage said in a Facebook video on Monday that he was willing to extend the government shutdown if the Legislature handed him a budget that included a tax increase. He also made clear he would sign off on a budget that removed the lodging tax hike.

“We are in very good financial shape, and there’s no reason why we should try to pass a tax when it’s not necessary,” LePage said.

Democrats, union activists and state workers spent the day calling for the shutdown to end and contended that it was hurting families and the state’s image.

“This is about getting a job done,” said Democratic Attorney General Janet Mills, a LePage foe, to a crowd of state workers. “Make the House Republicans do the right thing. There’s nothing to be debating anymore.”

The Legislature would need a two-thirds vote to pass the budget on its second vote. If the Legislature approves a budget, LePage would have 10 days to decide whether to sign or veto it.

LePage also created a stir on Monday when he suggested to several GOP senators that he was going to leave the state during the shutdown. Republican Sen. Roger Katz said LePage made clear he planned to leave “for 10 days.”

LePage spokesman Peter Steele told several media outlets that the claim is “fake news.” He didn’t return a message from The Associated Press seeking further comment.

Monday was the first day the shutdown was tangible for many residents, because it began over the weekend. Residents could not renew driver’s licenses, obtain vehicle titles and registrations or take road tests for new licenses. Numerous other government functions were similarly limited. Essential personnel such as a state police remained on the job.

Union members held a rally outside the State House on Monday morning to call for budget passage. Ray Heathco, an Auburn social worker, said his inability to do his job during the shutdown means child support enforcement cases can’t go forward.

“The cases are just going to sit there languishing,” Heathco said. “It’s really about support for children.”

All the budget proposals floated over the past several days have been in the area of $7 billion. Line items remained the sticking points.

The new budget proposal included $162 million in increased education spending to partially offset removal of a voter-approved 3 percent surcharge on the income of high-earning residents. The tax, approved in November, was designed to boost state spending on K-12 education.

Some Republican lawmakers said Monday they agreed with LePage’s stance on rejecting any budget that raised taxes.

“I know we can do better. We can say no to new taxes,” said Rep. Heather Sirocki, R-Scarborough, from the House floor.