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 surplus.