ST. PAUL — Republican leaders agreed Monday to meet Gov. Mark Dayton’s funding levels for Minnesota schools, courts and public safety in the next two-year budget, but they refused to increase their overall spending cap of $34 billion, leaving the two sides still far apart with the threat of a state government shutdown in less than a month.

After an hourlong meeting with GOP leaders, the Democratic governor said he appreciated what he called a “meaningful offer” but said he still wouldn’t support a budget without some new revenue. Dayton wants an additional $1.8 billion from taxes on the state’s highest earners, which Republicans have said they will not support.

“They’re still holding at $34 billion if I read between the lines, and again, that’s something that I will not agree to,” Dayton said. He has said it isn’t enough to prevent deep spending reductions to medical assistance programs, public universities, local government aid and other state programs.

Republicans said their latest offer was a legitimate attempt at compromise, noting that meeting the governor’s budget proposals in K-12 education, public safety and the judiciary would amount to $110 million more than they initially wanted to spend in those specific areas.

“This is a significant compromise from our caucus, from our membership,” said House Speaker Kurt Zellers, R-Maple Grove. “This is nearly 50 percent of the state’s total budget.”

But the $110 million would have to be swapped for reductions elsewhere. Where the cuts would be made to accommodate the money shift wasn’t specified in the new offer. Republicans said that would be part of ongoing negotiations with the governor.

In addition, Republican leaders said they told Dayton that if they met his requested K-12 spending level, he would have to agree to Republican-sought education reforms that he’s previously criticized.

Among those reforms is the elimination of state aid for school integration. Dayton vetoed a GOP-backed education bill last month that would have cut the funding, saying it “wrongfully harms poor children and children of color, which I will not accept.”

Dayton and Republicans have been divided for weeks in their budget negotiations. Dayton wants both sides to agree on a total spending figure, somewhere between the GOP’s $34 billion and his most recent offer of $37.6 billion, before he considers any specific funding cuts or additions.

While the state is projected to collect $34 billion in the next two years, it is also projected to have spending obligations that would total $39 million if spending were to remain at current levels. The difference between those two figures has resulted in the state’s $5 billion budget shortfall, but Republicans have insisted that aligning spending with available revenue is the best approach as the state recovers from the recession.

Republicans have tried to get Dayton to negotiate on separate budget bills, and their Monday offer amounted to another try.

Dayton appeared to give a small amount of ground, saying he was “slightly more optimistic” after Monday’s meeting and that would be willing to negotiate some specifics of different parts of the budget. However, Dayton said he saw little hope for resolution unless Republicans budge on their $34 billion cap.

And Dayton said likely wouldn’t call lawmakers into special session until a total spending figured is agreed upon, which could lead to a state government shutdown until an agreement is negotiated.

The last day of the current two-year budget is June 30. Failure to have the next budget in place by July 1 would mean a shutdown — likely stopping paychecks for tens of thousands of state workers, closing state parks and other facilities and numerous other service interruptions.

Members of Dayton’s staff and Republican committee chairs are scheduled to meet Tuesday to discuss transportation and judiciary budgets. Dayton and GOP leaders are due to meet again on Wednesday.

On Friday, about 35,000 state workers are scheduled to receive layoff notices.