ST. PAUL — A sweeping shutdown of Minnesota’s state government appeared certain as a midnight deadline approached, with Democratic Gov. Mark Dayton and top Republicans breaking off negotiations to accuse the other side of stubbornness.

Republicans said a deal was close and pleaded with Dayton to agree to a bare-bones budget bill to avert a shutdown while they come back in special session next week to wrap things up.

Dayton rejected it as a publicity stunt, saying the two sides aren’t close at all, and called on Republicans again to agree to hike taxes on top earners. Asked whether there was any way to stop a shutdown, Dayton said: “I’m not aware of it.”

A shutdown would force thousands of layoffs, bring road projects to a standstill and close state parks just ahead of the Fourth of July weekend. The effects were already being felt hours ahead of the deadline, as people rushed Thursday to get driver’s and fishing licenses, and park officials began warning campers to pack their gear and leave.

Though nearly all states are having severe budget problems this year, Minnesota stood alone on the brink of a shutdown, thanks to Democratic Gov. Mark Dayton’s determination to raise taxes on high-earners to close a $5 billion deficit and the Republican Legislature’s insistance that the gap should be closed by cutting spending.

Negotiations between Dayton and legislative leaders were fitful, starting and stopping with no outward signs of progress, and details were scant, since the two sides agreed to what they jokingly called “the cone of silence.”

Republican Sen. Michelle Benson said she wasn’t budging.

“If we don’t start taking a different approach to how we manage our government, we’re going to swing from one bad economic circumstance to another,” Benson said. “We can’t just keep throwing more money at government and hoping that makes things better.”

The showdown was something of a small-stage version of the drama taking shape in Washington between President Barack Obama and the Republicans over taxes and the nation’s debt ceiling.

Though many states are having budget difficulties this year, those where political power is concentrated in a single party easily passed budgets. Some of those with divided government had healthy reserves, including Alaska, Iowa and Montana; Minnesota’s rainy-day accounts are drained. Others such as Louisiana and Nevada used one-time money or federal dollars to patch things together. Nevada and Missouri renewed taxes.

In New Jersey, Republican Gov. Chris Christie used the line-item veto Thursday to pare a budget from the Democratic-controlled Legislature before signing it into law, preventing a shutdown.

Only four other states — Michigan, New Jersey, Pennsylvania and Tennessee — have had shutdowns in the past decade, some lasting mere hours.

A stoppage in Minnesota would halt non-emergency road construction, shut the state zoo and Capitol, and stop child-care assistance for the poor. More than 40 state boards and agencies would go dark. Critical services, including the State Patrol, prisons, disaster response and federally funded health, welfare and food stamp programs, would not be affected.

State park officials told campers to strike their tents well before the deadline, even though there was still a chance of a deal. They said it would be too difficult to herd campers out in the middle of the night if talks failed.

In Afton State Park, near St. Paul, Rick Miller of Elko-New Market pushed up a camping trip with his 7-year-old son, Jack, to beat the shutdown. Miller originally hoped they could spend Thursday and Friday nights in the park on the picturesque St. Croix River, but he booked a campsite for Wednesday night.

“With the shutdown we decided we better come and get it in,” he said. “We don’t know how long it will be before we can get back into a state park.” He added: “It’s too bad they can’t just get the job done.”

A small group of protesters paraded before reporters clustered outside Dayton’s office on Thursday afternoon, chanting and waving signs to support the governor’s position. “You say cut back, we say fight back!” they yelled. One woman carried a handmade sign that read: “GOV DAYTON DON’T BACK DOWN!”

Dayton is Minnesota’s first Democratic governor in 20 years, and Republicans are running the entire Legislature for the first time in 38 years.

Dayton’s predecessor as governor, Republican presidential hopeful Tim Pawlenty, took time away from campaigning Thursday to hold a news conference at the Minneapolis-St. Paul Airport, where he urged legislative Republicans to hold their ground.

“This country needs to get its government finances under control,” said Pawlenty, whose leadership many Minnesota Democrats blame for the state’s budget problems. “That needs to happen in Washington, D.C., and that needs to happen in St. Paul, Minnesota.”

Dayton has proposed raising taxes on couples earning more than $300,000 and individuals making more than $180,000. Republicans have opposed any new taxes or new revenue sources, arguing instead that the state should rely on spending cuts, including deeper reductions in health and welfare spending than Dayton is willing to accept.

Some GOP moderates have talked of breaking the impasse with other means of raising revenue, such as eliminating tax breaks or authorizing a casino. Dayton has said he is open to such ideas.

Rank-and-file Republicans gathered at the Capitol on Thursday, more than a month after their regular session ended. Members of the large Republican freshman class, whose election victories in November helped the party take control of the Legislature for the first time in decades, held tight to their message that a total two-year state budget of $34 billion is big enough.

“I personally think the Republicans will probably be more damaged than the governor” by a shutdown, said freshman Rep. Mike LeMieur, R-Little Falls, who toppled an incumbent Democrat in November. “The fact is that we’re all up for re-election again next year, and he’s not up for three years.”