Gov. Mark Dayton said Monday that he and Senate Democrats had agreed on a K-12 school funding increase that is double what Dayton originally called for, opening an even wider gulf between Democrats and House Republicans in the final week of the legislative session.

Dayton upped his school spending demands after a private meeting with top DFL lawmakers. Later in the evening, Dayton and leading lawmakers from both parties resumed their negotiations, trying to bridge differences on major pieces of the two-year budget.

After about three hours of talks at the governor's residence focused primarily on the Health and Human Services budget, they broke without making any comment to reporters. They're expected to reconvene Tuesday afternoon to talk education.

Republican House Speaker Kurt Daudt expressed frustration late Monday that talks were cut off early. He said he was told negotiations would break for dinner, and anticipated them continuing until midnight. The governor's office said education discussions would be postponed because Education Commissioner Brenda Casselius was unavailable.

"If that didn't work and they needed her there," Daudt said, "there are six other things we could have talked about tonight, including continuing the transportation discussion."

Earlier discussions on transportation were largely fruitless, Daudt said, because Dayton and Bakk wouldn't budge from their proposed gas tax increase, which he said is a non-starter in the House.

Republican House Speaker Kurt Daudt criticized Democrats for seemingly moving even further from the GOP on school spending levels.

"I actually think this could all come together very quickly," Daudt said. "But unfortunately, the other side seems to be moving farther away, not closer to where we are, so that makes it more difficult."

Dayton initially called for a 1 percent increase in school funding for 2016 and 2017. The Senate approved that same amount, which would boost per pupil payments to school districts by $57 per student per year.

But on Monday, Dayton said he now wants 2 percent. That's after some school districts have complained that anything less than a 1.5 percent increase per year would result in teacher layoffs, larger class sizes and other program cuts.

Rep. Jenifer Loon, chief sponsor of the House education bill, said it would be hard to provide that much aid while also delivering Dayton's universal preschool initiative.

"I'd love to do it, but we have to get the budget to balance with the resources available," said Loon, R-Eden Prairie.

The GOP-led House has approved a 0.5 percent increase per year in its education funding bill.

State aid dollars form the basis of most school district budgets.

"The Senate and I agree that 2 percent … for each of the next two years is the minimum of what's necessary," Dayton told reporters. The new gap means a greater bridge to cross as Dayton and legislative leaders work to figure out the next two years of state spending, which includes complicated discussions about tax rates, transportation spending and other pieces of the massive state budget.

Dayton added he is not surrendering his other high-profile education priority, a funding boost of about $350 million aimed at helping every public school statewide offer preschool for 4-year-olds.

And while Dayton characterized the 2 percent annual boost for two years as agreed upon between himself and DFL senators, Majority Leader Tom Bakk left some doubt.

"I'm not suggesting it," said Bakk, DFL-Cook. He said Dayton brought up the proposal during Monday's meeting, but did not indicate where money in the budget could be shifted to cover that kind of additional spending.

The Senate's budget blueprint would leave $250 million in a reserve fund.

Bakk said that money conceivably could be rerouted to cover a larger aid increase to schools, but said that could come at the expense of other major session priorities.

"I was hoping to save it for a deal on taxes and transportation," Bakk said.

This kind of high-level trading on seemingly unrelated issues is common to the final week of a legislative session, as the governor and lawmakers work toward a final spending plan that satisfies their often wildly divergent priorities.

House Republicans are seeking more than $2 billion in tax cuts, just a little more than the size of the state's projected budget surplus. Both DFLers and Republicans say they want major new investments to upgrade the state's transportation infrastructure, but differ on how and how much. And while Dayton calls the preschool funding proposal his top priority, the funds needed are not currently included in either House or Senate school spending plans. The two sides also remain far apart in total spending levels for health and human services programs.

The deadline to adjourn the regular session is May 18 at midnight.

Abby Simons contributed to this report.