ST. PAUL — Gov. Mark Dayton and Senate Democratic leaders upped the ante Monday on school spending as top lawmakers met in private to craft a new state budget, although a deal appears no closer than a week ago.

Dayton said the Democratic allies, who are deep in talks with a GOP House, are coalescing around a 2 percent bump to the state’s per-pupil funding formula for K-12 schools for each of the next two years — more than double what was passed in the House GOP budget. Dayton said that additional funding shouldn’t preclude additional money for statewide preschool, one of his top priorities. House Republicans said they could get on board with an increase, so long as reductions were made elsewhere to account for the $170 million proposition.

Dayton and legislative negotiators have reported little substantial progress in private discussions that ranged from closed sessions in state offices to a fishing boat on Lake Vermillion. The gap between the two parties on the size and scope of proposed tax relief, health care spending cuts, school funding and money for road and bridge repairs all threaten to push the Legislature past its May 18 adjournment and into overtime.

Negotiation sessions shifted to the governor’s residence Monday evening to ensure any deal will have Dayton’s approval, but those discussions broke off after a few hours, to the frustration of House Speaker Kurt Daudt.

Standing in a nighttime drizzle outside the mansion gates, Daudt was openly agitated about the lack of real movement. He said days of talks had resulted in $20 million of progress in a two-year budget exceeding $40 billion.

“At this pace it’s going to take us, I don’t know, 20 years to get to where we need to be,” Daudt said, adding that a Democratic-backed gas tax remained a major hang-up.

Earlier, Senate Majority Leader Tom Bakk acknowledged the time window for a deal was narrowing.

“We’re getting to the point where we just have to recognize some things are going to have to fall on the floor,” Bakk said. “We may end up in that situation by the end of the week.”

The pressure is on to get a deal and avoid a third partial government shutdown in the last decade. Dayton said that his administration would have to send out layoff notices to state workers by June 1 to prepare for a possible shutdown in July. This time, a nearly $1.9 billion surplus gave lawmakers extra breathing room — but also more to argue about.

Bolstered by calls from their own party to “give it back,” House Republicans lined up more than $2 billion in tax relief through two years of income tax exemptions, business property tax cuts and a gradual phase-out of Social Security income from state taxes. Daudt said he knows the final budget won’t go that far, and on Monday urged Democrats to compromise with the House on their own priorities.

But Dayton said he wouldn’t accept even $1 billion in tax cuts.

“It sounds good, it’s a nice re-election ploy but it’s not fiscally responsible,” he said.

In one of the sharpest departures with the Senate, the new GOP House majority aimed to cull more than $1 billion from the state’s mammoth health and human services budget by ending a low-income health care program and weeding out ineligible enrollees from public programs. Dayton declared MinnesotaCare “off the table absolutely this year” but said he is open to a task force to study future retooling.

The waste and fraud savings themselves have become a flashpoint between the two parties. Republicans peg the savings at $300 million or more, but Democrats point to the state’s own fiscal analysis suggesting it’s just $17 million over the next two years.

Meanwhile, Bakk is still hopeful he can pass his top prize: a long-term, multibillion-dollar transportation package funded by a dedicated gas tax hike. But the Cook Democrat stressed a transportation bill — much like a tax bill — isn’t necessary to pass a budget.

In an effort to slim the long list of areas they need to find agreement, Bakk said he offered to drop a transportation package if a tax bill was also off the table. Republicans declined.

“I don’t want that to be the outcome,” Bakk said. “They can’t expect that I’m going to give up on my No. 1 priority and they still continue to get theirs.”