ST. PAUL — Minnesota’s $5 billion deficit remained unresolved Thursday, even after a private meeting between Democratic Gov. Mark Dayton and rank-and-file Republican lawmakers who oppose his proposal to raise income taxes on the biggest earners.

Dayton and top GOP legislators emerged from the hourlong meeting to stand before cameras together and hold out hope for a budget deal by the Legislature’s Monday’s adjournment deadline. Dayton said he will meet with legislative negotiators Friday to discuss spending for public schools and state agencies. He didn’t repeat his threat to reject all budget bills without an overall budget deal.

“We’ll continue the dialogue and negotiations starting tomorrow morning,” Dayton said.

He added: “Our discussion was solely focused on right now.”

Behind closed doors, the first-term governor delivered a stern message to the GOP lawmakers who took over both legislative chambers for the first time in 38 years earlier this year. According to the prepared text of his remarks, he said they would usurp his powers by giving the legislative branch “the sole power to decide the state’s budget” and disenfranchise more than 919,000 voters who supported him.

“The crux of our current impasse is no longer policy, or ideology, or even budget. It is your unwillingness to assume the responsibilities of leadership,” Dayton said.

Republicans have dug in against raising taxes or spending more than the $34 billion the state is projected to collect over the next two years. Dayton’s latest offer would raise $1.8 billion in new taxes, mostly from a new top income tax tier for couples earning more than $300,000 and single filers with yearly incomes topping about $180,000.

Dayton and Democrats decry the Republican approach, saying cuts and spending reductions in their budget bills target some of the state’s most vulnerable communities and put pressure on city and county leaders to raise property taxes to make up for fewer resources from the state.

After the meeting, there was little sense among freshman GOP lawmakers that Dayton persuaded them to move off their position. Several said they were elected on pledges not to raise taxes.

“I ran on just two promises: I would not vote for a tax increase and I’d vote 100 percent pro-life,” said Rep. King Banaian, who won an open seat by 13 votes in a St. Cloud district previously represented by a Democrat. “If I violate either one of those pledges, I don’t deserve to be brought back at all.”

Sen. John Sterling Howe of Red Wing said he saw no chance his caucus would agree to spend more than $34 billion.

“We think we are in the middle already,” said Howe, who won an open seat that had long been represented by a Democrat.

GOP caucus leaders softened their message somewhat as they stood with Dayton. But they gave no indication that they intend to agree to his tax proposal or more state spending.

“There was some give and some take,” Senate Majority Leader Amy Koch said.

She said members told Dayton: “They want government to live within its means and they want government to hold the line on spending.”

The budget standoff threatens to push the session into overtime.

As House Republicans filed from the meeting room after their time with Dayton and headed for the House chamber, they were greeted by a throng of protesters under the impression the House would vote on a proposal that would ask voters in 2012 whether or not they want to add a ban on gay marriage to the state constitution.

Activists on both sides of the issue were chanting directly at each other and lawmakers, producing a roar that echoed throughout the Capitol and could at times be heard inside the House chamber. The House ultimately didn’t take up the amendment Thursday, and House leaders have been tight-lipped about when the vote would take place.

Several hours after the meeting between Dayton and lawmakers, the Legislature finished passing the $34 billion budget Republican budget. The Senate approved three last bills — spending packages for higher education, transportation and state parks. All the budget bills passed both the House and Senate with nearly unanimous opposition from legislative Democrats.

If there’s no budget deal by Monday, Dayton would have to call a special session. Parts of state government would shut down starting July 1 if a new two-year budget isn’t in place by the end of June.

One Republican veteran, Rep. Steve Smith of Mound, said he saw a need for not just Dayton but also members of his own party to show more flexibility.

“Maybe it’s time for Republican leadership to understand they’ve got to get their feet out of the cement, and the governor has to get his out of the cement, and they need to start moving toward each other,” Smith said. “My advice is both sides ought to get their feet out of concrete.”