ST. PAUL — A months-long dispute between Gov. Mark Dayton and legislative Republicans briefly shifted Wednesday from taxes and spending toward the size and scope of a government shutdown made increasingly likely by their inability to agree on a state budget.

The shutdown squabbling broke out after Dayton filed a petition in Ramsey County court laying out a bare-bones plan for maintaining critical state services if the government shuts down July 1. Legislative Republicans, for months insistent on greater austerity in state spending, found themselves questioning the severity of the Democratic governor’s approach to a shutdown.

Dayton, who said the petition was submitted “with a heavy heart,” asked the court to keep about 13,000 of 36,000 state employees on the job to deliver programs vital for public safety and health. The administration’s plan recommended closing 46 state boards and agencies entirely and keeping minimal staffing levels at 29 other agencies, with the highest number of employees retained at the departments of Human Services, Corrections, Public Safety and Veterans Affairs. State parks would close, as would offices that provide various state licenses.

“Difficult as this would be, a shutdown, it still pales in comparison to the devastating effect of the Republican budget on critical areas,” said Dayton, whose petition also asked the court to appoint a mediator to facilitate budget negotiations. “A temporary shutdown, painful as it would be, is not equivalent to the kind of catastrophes that would be ongoing that would occur if I acceded to this budget. That’s just a fundamental principle.”

At a meeting of a legislative budget panel Wednesday, Senate Majority Leader Amy Koch, R-Buffalo, suggested Dayton’s case for what wouldn’t be funded in a shutdown was “fairly broad.” She and fellow Republicans suggested the governor was more limited in his definition of critical services than were previous administrations as they prepared for a 2001 shutdown that was averted at the last minute, and a short shutdown in 2005.

“He has said all along that he would not accept our bills because they didn’t fund enough higher education, special education, seniors, disabled and health care and yet he is willing to impose this kind of budget and willing to force those folks into a shutdown that would cut off the very services he claims they need,” Sen. Julianne Ortman, R-Chanhassen, said at the Legislative Committee on Planning and Fiscal Policy meeting. “So I would ask you, governor, whose budget is more draconian?”

Among the state services Dayton said should not continue in a shutdown were aid payments to public schools; payments to health care providers and vendors that serve recipients of state assistance programs; and transportation programs including highway construction projects, bridge assessments, and operation of ramp meters.

Under Dayton’s petition, many noticeable functions of state government would be halted. The Department of Natural Resources would maintain 220 employees for conservation law enforcement, dam and water control safety monitoring and a handful of other operations, but not to keep state parks operating. Agencies including the State Lottery, the state Arts Board and the office of Administrative Hearings would close entirely, as would numerous licensing boards for occupations from physical therapy to optometry to private detectives.

Dayton’s court petition argues that five statewide objectives must still be met in a government shutdown: basic custodial care for residents of prisons, treatment centers, nursing and veterans’ homes and residential academies; maintenance of public safety and immediate public health concerns; payment of government benefits to individuals; preservation of the essential elements of state government’s financial structure; and preservation of the necessary administrative and support services to provide the above provisions.

The government sent layoff notices to tens of thousands of state employees last week, and the state Department of Human Services mailed notices to 572,000 clients Tuesday warning that they might lose access to health care assistance and other services if the Minnesota government shuts down.

In the filing, Dayton suggested former state Supreme Court justices Kathleen Blatz and James Gilbert — both of whom were appointed by Republican governors — as mediators “to oversee and facilitate negotiations between the legislative majority, on the one hand, and the legislative minority and the governor, on the other.”

Attorney General Lori Swanson had already suggested Gilbert serve as a court-appointed “special master” who will ultimately decide which state services be deemed critical. Dayton’s court filing indicates that the special master’s duties should include mediation.

The court filing says Dayton “commits to be present at the mediation and to devote his full time and attention to reaching an agreement.”

Dayton, Koch and House Speaker Kurt Zellers met briefly Wednesday afternoon but said afterward they talked more about the process of budget negotiations in the coming days than actual negotiating on spending levels. Dayton and the leaders planned to meet again Thursday.