ST. PAUL — Minnesota’s legislative leaders on Tuesday started to chip away at the massive differences that threaten to hold up a new, two-year budget, with Gov. Mark Dayton offering to slash the amount of spending he wants and agree to more in tax relief to strike a deal with Republicans.

The Legislature has less than a week to finalize a budget, and it’s not a done deal yet. Aiming for more than $1 billion in tax breaks, Republican leaders said they’re looking for more in tax relief than the $500 million Dayton offered.

But in just an hour-long private meeting, the two sides started to close a $1 billion gap between them heading into the homestretch of the session — Dayton has previously proposed a roughly $46 billion budget, while Republicans’ proposal clocks in at nearly $45 billion. All told, Dayton offered to cut his spending requests by more than $624 million, with unspecified reductions to state agencies, jobs and economic development programs and funding for courts and public safety.

House Speaker Kurt Daudt called it “significant movement” — but not without some steps backward.

Critically, Dayton proposed raising license tab fees as part of a broad transportation funding package that has been more than three years in the works. Dayton’s proposal would raise $321 million for road and bridge repairs, though his administration couldn’t say Tuesday what that would mean for the average driver.

Raising license tab fees was part of an emerging deal last year between Democrats and Republicans, when the GOP controlled the House but not the Senate. But Republicans’ takeover of the Senate in November has given the GOP extra power to stand firm against any tax or fee increases that Dayton has deemed necessary.

Daudt and Senate Majority Leader Paul Gazelka called its renewal a setback and unacceptable.

“We have felt like it was off the table for some time,” Daudt said.

Republicans planned to make their own counter-offer later Tuesday afternoon.

The offer-trading follows more than a week of standstill at the Capitol, after the GOP backed away from the negotiating table and sent Gov. Mark Dayton their own budget bills. Dayton vetoed all 10 such bills.

Despite the return to talks, legislative leaders are only talking in round numbers and not about specific changes to programs that could affect Minnesotans. That means the fate of Dayton’s prized preschool program — which Dayton sought to expand but Republicans proposed eliminating — is still up in the air. So are the specific tax breaks that Minnesota residents could get from Dayton’s $500 million offer.

Republicans signaled they’ll still push for even more in tax relief.

“It’s not where we want to be on that number, but that’s a big step,” Gazelka said.