ST. PAUL — The next-to-last day of Minnesota’s legislative session trickled away Sunday with no evident movement on a budget impasse between Gov. Mark Dayton and Republican legislative leaders.

The two sides are about $1.8 billion apart in how to resolve a $5 billion deficit, and if they don’t strike a deal by midnight Monday, a special session would be necessary to avert a government shutdown later this summer.

“Nothing to report big at this point, but discussions are continuing,” said Senate Majority Leader Amy Koch, R-Buffalo, after leaving an hour-long meeting with Dayton and other legislative leaders in his office.

Republicans passed a set of budget bills that total $34 billion in spending; the Democratic governor wants to supplement that with another $1.8 billion in new revenue in order to avoid some of the steepest cuts and reductions to social services, state aid to cities and counties and spending on state colleges and universities.

Dayton suggested the extra revenue come from an income tax increase for those in the state’s highest bracket of earners, but has said he’s open to other ways of raising state revenue. Earlier in the week the governor halved the amount of new revenue he’d like to see, but Republicans have refused to budge from the $34 billion figure.

“If we raise taxes during this economic recovery, we’ll squelch whatever growth we had and we’ll be right back where we were for the next two years,” said House Speaker Kurt Zellers, R-Maple Grove.

Zellers said he and other Republican leaders were willing to work day and night with Dayton to reach an agreement. But there seemed little urgency to their meetings: when Zellers, Koch and their colleagues left Dayton’s office about 3 p.m., they said they’d probably meet again later in the day but that nothing was scheduled.

Dayton had not appeared from behind closed doors by late afternoon. He stayed in his office meeting with Republican chairs of various legislative finance committees to discuss concerns he had with their budget bills, but reports from inside those meetings brought no further hints of progress.

“The governor repeated what he’s been saying, that he wants to solve the budget with a global solution,” said Rep. Bud Nornes, R-Fergus Falls, the chairman of the House Higher Education Finance Committee, after meeting with Dayton. “That didn’t leave us with much else to talk about. I’m not sure the governor’s been very clear, besides saying he wants a tax increase, about what might be some other ways of solving this thing.”

Both the House and Senate were meeting in late afternoon floor sessions to work on policy bills. The night before, the House passed a constitutional amendment that will go to voters in 2012 asking whether they want to define marriage in the state constitution as between a man and a woman only.

Another issue that got a lot of attention at the Capitol in recent weeks, the push by the Minnesota Vikings to secure partial taxpayer financing for a new football stadium, had little heat around it in the final days. It also appeared destined to be the subject of continued negotiations before and during a special session.

If midnight Monday hits without a budget deal, only Dayton can call lawmakers back for a special session. But whenever he does, lawmakers can stay in session as long as they want.