ST. PAUL — Picking up where the legislative session ended, Gov. Mark Dayton and House Speaker Kurt Daudt began their search Tuesday for a way out of Minnesota’s latest budget impasse that has made a special session a certainty and a partial government shutdown a possibility.

The Democratic governor and Republican legislative leader met for an hour with a promise to reconvene today. Afterward, they voiced a joint desire for a swift resolution for reviving three vetoed budget bills and two other measures lawmakers failed to pass in time. Dayton was to meet later with Democratic Senate Majority Leader Tom Bakk about the budget bills and other items considered unfinished during the regular legislative session.

Late last week, Dayton struck down bills covering public school funding, agriculture and natural resources programs and economic development programs. He considered the school spending too meager and took issue with some major policy changes in the other two. Dayton issued a list of more than a dozen “special session priorities” for adding provisions to or subtracting them from the bills he vetoed.

“We want to be flexible and we want to get the work done that Minnesotans expect us to get done,” Daudt said. “If that means we have to give a little bit, we’ll have to do that as well.”

Daudt said Republicans might ask for the inclusion of some of their priorities, such as teacher tenure law changes they advanced earlier this year.

Lawmakers have until July 1 to pass replacement bills or some state agencies would go dark, idling up to 10,000 public employees in the process. Dayton said he wants an agreement in place before calling a special session.

Minnesota has had two partial government shutdowns in the last decade, the most recent in 2011. A judge was brought in to declare services essential and keep money flowing to those programs, but thousands of state employees were off the job for weeks.

“June 30th is a huge cliff that I don’t have any desire to go over again,” Dayton said, noting that layoff notices would go out next week to prepare for a wind-down of the affected agencies.

Daudt and Dayton did reach agreement on one score: With the state Capitol closed for renovations, they said the special session would be held in the adjacent State Office Building that is already wired for television airing and close to legislative staff.

Dayton is holding out for a substantial amount of new education spending — up to $650 million more than the state commits now — with a chunk of that devoted to quality pre-kindergarten programs for 4-year-olds. He vetoed a bill with $400 million in new public school spending, arguing it was insufficient.

Dayton has floated the idea of pairing $250 million in extra school spending with $250 million in middle-class tax cuts, modeling them after a Republican plan that stalled out.

Daudt didn’t immediately embrace putting adding a tax plan to the special session mix.

“Every time we bring an extra bill in, it opens up a whole bunch of other unanswered questions,” he said. “If we try to keep this simple we probably wouldn’t bring that in. We’re certainly open to that.”

As for an environment and agriculture budget bill, Dayton took issue with borrowing from a closed landfill remediation account and for shifting some regulatory review authority away from the Minnesota Pollution Control Agency and its citizen board. That bill helps pay for Minnesota’s avian flu response and upkeep of state parks, making swift passage of a substitute crucial for all sides.