ST. PAUL — A Minnesota judge is considering whether Gov. Mark Dayton’s veto of funding for operation of the Legislature was unconstitutional as a bitter legal dispute headed for its first court appearance Monday.

The fight over funding stems from a messy budget-setting process that required a special session to finish up. Dayton signed much of the $46 billion spending package but zeroed out funding for the Legislature itself, trying to force lawmakers to rework costly provisions of a $650 million tax bill and other measures. Republicans who control the Legislature showed little interest in renegotiating and filed a lawsuit.

Just last week, the two sides agreed to temporarily fund the Legislature through October to avoid any layoffs or service cuts that would begin July 1. But even that will hinge on Ramsey County Court Chief Judge John Guthmann’s approval, and attorneys for both sides said they’d need a court’s direction to find a resolution.

“We are at an impasse,” said Sam Hanson, Dayton’s attorney in the case.

House Speaker Kurt Daudt said Monday he hoped for a decision later this week. He said he thought Dayton shouldn’t appeal if Guthmann sides with the Legislature, but vowed to appeal “absolutely” if they lose the case.

State government officials packed inside the Ramsey County courthouse as their dispute formally entered the third branch of government. The case could have broad implications, both for the working relationships between the GOP Legislature and Dayton in his final months in office as well as major court precedents for how the executive and legislative branches interact.

Doug Kelley, the Legislature’s attorney, asked the court to void the governor’s line-item veto and restore its roughly $130 million in funding. Kelley called it a clear violation of the separation of powers that would “obliterate” the House and Senate, arguing Dayton’s line-item veto power should be limited to specific appropriations he dislikes and not as a means to gain leverage.

“It’s clear to us that he’s not objecting to $130 million in funding for the Legislature,” Kelley said. “We are not going to negotiate with a gun to our head.”

But Guthmann worried that Kelley’s view would curb a future governor’s power to rein in legislative overspending.

Hanson argued that the governor’s authority to veto even funding for legislative or judicial branches is absolute, however “distasteful” those decisions could be. And he rejected Guthmann’s suggestion that the two sides use a mediator to find a political solution.

“I don’t see it happening while this legal question is unresolved,” Hanson said.

Taxpayers are footing both sides of the bill: Kelley’s team charges the Legislature up to $325 hourly, while Hanson charges more than $506 an hour. Both attorneys offered discounts in the case.

And that’s not the only case involving the Legislature. Guthmann also heard initial arguments in a separate lawsuit seeking to force $14,000 raises on lawmakers despite lawmakers’ decision not to appropriate money to cover their first pay increase since 1999.