ST. PAUL — Republican leaders of Minnesota’s Legislature sued Gov. Mark Dayton on Tuesday, arguing his veto of their funding in a dispute over tax breaks is unconstitutional.

The lawsuit was in the works for weeks, prompted by Dayton’s line-item veto on May 30 that nixed all funding for both the House and the Senate — nearly $130 million in all. Republicans immediately cried foul, calling it a blatant breach of the separation of powers and vowing to challenge the governor in court.

The GOP finally filed Tuesday after a final meeting with Dayton that solved nothing.

It’s not clear how quickly the dispute may be resolved. It could wind through the state courts or be accelerated directly to the Minnesota Supreme Court. And the clock is ticking: state funding for the legislative branch will dry up July 1, when the state’s new budget kicks in.

House Speaker Kurt Daudt said he hopes it won’t take a court resolution. But he said Republicans won’t renegotiate bills already signed.

Dayton said House and Senate leaders have the ability to resolve the matter before funding runs out.

“They’re sticking with their position, and I’m sticking with mine,” Dayton said.

The legal dispute stems from end-of-session political maneuvering as Dayton and lawmakers sought to wrap up a $46 billion budget. Republicans forced the Democratic governor’s hand into signing a package of $650 million in tax breaks, linking funds for the state’s 1,300 Department of Revenue employees. Dayton responded by vetoing funding for the Legislature itself, seeing it as leverage to renegotiate several tax measures he deemed too costly, like tax breaks for wealthy estate owners, commercial property taxes and some tobacco products.

The vetoed funding pays for items ranging from salaries — the Legislature has more than 200 legislators, with more than 400 full-time staffers to support them — to expenses to building lease payments.

The Legislature finalized a contract with Minneapolis firm Kelley, Wolter and Scott last week that calls for charges of up to $325 an hour for its top attorney’s services, with lower rates for others. The firm agreed to a 50 percent discount for the contract, which calls for one lump-sum payment once the case concludes.

Dayton has singled out tax breaks for cigarettes and premium cigars as particularly harmful, saying that freezing current cigarette taxes at $3.04 a pack by removing future automatic increases and slashing taxes on top-dollar cigars from $3.50 each to just 50 cents would reverse recent decreases in youth smoking rates. Dayton also wants lawmakers to retool changes to how teachers are licensed and remove a duplicative rulemaking ban on expanding driver’s license access to immigrants living in the state illegally.

But Republicans have insisted it’s too late to rework those provisions, arguing that Dayton should have vetoed those bills if he didn’t like them. Those issues were large areas of dispute in the debate over a $46 billion budget that Dayton signed into law late last month.