ST. PAUL — Hoping to avoid a Minnesota government shutdown next month, top Republican lawmakers mapped out a series of June budget talks on Friday but made no move toward Democratic Gov. Mark Dayton in their dispute over taxes and spending.

The two sides spent nearly 90 minutes together at the governor’s St. Paul residence in their first private meeting since the Legislature adjourned last week without a budget deal. They plan to meet again Monday, when Dayton said he expects legislative leaders to spell out whether they will agree to more spending, and how much.

“If they’re not willing to meet me halfway, if they’re going to stand on $34 billion, then the rest of this is just for show,” Dayton said after the meeting. “I guess we’ll find out next week.”

Majority Republicans insist on capping spending at $34 billion, the amount projected to come into state coffers over the next two years. Dayton wants to add $1.8 billion in new taxes, mostly from the highest earners, to soften cuts to projected spending on public health care programs, universities, local aid and other services. He cut down his proposal for new taxes last month from more than $3 billion.

The governor said he would consider a “midpoint” of spending to be somewhere around $35.5 billion to $35.8 billion — below the projected $39 billion if spending continued at levels set in current law, and less than his opening offer of $37.6 billion.

Asked about spending more than $34 billion, House Speaker Kurt Zellers made it clear the GOP position hasn’t changed.

“You cannot spend above what you have — a pretty simple concept that most Minnesotans and their family budgets get,” he said.

Senate Majority Leader Amy Koch described the meeting schedule as a “roadmap” to prevent a shutdown. It would have Dayton and legislative leaders meeting every weekday through June 23. It calls for finalizing a budget agreement June 21 and starting a special session June 27.

Only Dayton has the authority to call a special session, and he repeated Friday that he won’t do so without an overall budget deal. Still, the governor didn’t reject the GOP schedule outright, saying he was “perfectly willing” to meet, but wants the meetings to “lead somewhere.”

Failure to enact a new budget by the end of the month would result in a wide-ranging government shutdown starting July 1, likely closing state parks, driver’s license renewal counters, highway rest areas and other services dependent on state dollars.