ST. PAUL — Republicans in the Minnesota House offered Tuesday to hike license tab fees on new cars, the first time in the contentious debate over the transportation budget that they’ve agreed to raise new money to help fund $600 million in needed annual road and bridge repairs.

It’s just the latest installment in the Legislature’s back-and-forth battle to piece together a transportation package. GOP lawmakers’ proposal to hike tab fees — from just a few dollars more on brand new cars to nearly doubling costs on older models — follows Gov. Mark Dayton’s own, larger tab increase offer in lieu of the gas tax increase he and Democrats have continually sought.

Their plan also would tap a budget surplus and borrow money for construction projects.

But with less than a week ago, there’s still plenty of ground to cover. The GOP’s plan notably left out funding for mass transit projects, such as a light-rail train line from Minneapolis to southwestern suburbs. House Speaker Kurt Daudt said a plan with funding for the light-rail project wouldn’t earn a single vote among the 73 Republicans who control the House.

“It’s not easy. We’re not just $20 million apart here and there,” the speaker said. “I feel like we need to break the ice and get to agreement on something.”

Dayton and legislative leaders emerged from several hours of private meetings Tuesday and reported some progress but no firm agreements. They’ve put their focus on transportation talks as a linchpin for broader deals on how to spend a $900 million budget surplus.

Senate Majority Leader Tom Bakk keyed in on the divide over transit projects as a major stumbling block.

“The Senate has been clear that there’s not a transportation deal that doesn’t include transit,” the Cook Democrat said.

The bulk of the GOP’s plan would be paid for with $300 million from the state’s general fund. It also calls for borrowing $200 million each year for construction projects — a figure that Dayton argued shouldn’t count because the state already borrows that money — and raise another $100 million by increasing the state’s annual license tab fees on both brand new vehicles and newly purchased used cars.

A driver with a five-year-old car worth $15,000 would pay $146 under the GOP’s plan, compared to the current rate of $123. The system levies larger hikes on older cars: The annual bill for a 10-year-old car would nearly double, from $48 to $90. Owners of 11-year-old cars or older would see a slight decrease.

The increases Dayton outlined in his own offer Monday were far larger: The payment on the same, five-year-old car would rise to $203. The Democratic governor’s proposal was one of two attempts at compromise, one of which also included a 5-cent gasoline tax increase.

While session formally ends next Monday, a constitutional ban on passing bills in the final 24 hours pushes the effective deadline back to Sunday. And there’s another time pressure too: Legislative leaders have been told they need to send bill language to the state revisor’s office by Thursday to get drafted for final votes.

“Sometimes it takes a deadline to get people to move from their position,” Daudt said.