AUGUSTA — Cities and towns across Maine will be watching Augusta this week, where lawmakers will consider legislation allowing them to redo their school districts’ budgets to spend unexpected state aid — without having to open the polls for voter approval.

The measure is among the final bills the Democratic-controlled Legislature will take up when it reconvenes on Tuesday, along with a number of bills that have been vetoed by Republican Gov. Paul LePage that require a two-thirds vote for an override.

Critics of the bill say voters should have a say in how the extra money is spent. But supporters say that holding another school budget vote will be financially burdensome for cities and towns.

“I still think that we retain the democratic process,” said Sen. Rebecca Millett, a Democrat from Cape Elizabeth and the bill’s sponsor. “It’s just extraordinary situation, so we’re trying to provide them with a little bit more flexibility.”

The late passage of the $6.3 billion, two-year state budget means that many school districts approved their budgets before they knew how much funding to expect from the state.

With an extra $28 million going to schools this year, some districts will get more than $1 million than originally thought. At the same time, the budgets of some districts — including those serving Portland and Bangor — don’t account for additional retirement costs that the state once picked up.

Current law says school districts cannot use that extra money unless they redo the spending plans and ask residents for approval. But Portland Mayor Michael Brennan and others are asking for a one-time reprieve of the voter validation.

Portland Public Schools will receive $1.9 million more next school year than what had been proposed in LePage’s budget. It also will have to pick up $1.3 million in teacher retirement costs that weren’t budgeted for. Brennan said reopening the polls will cost Portland $12,000-$15,000, hurting the already cash-strapped city facing revenue sharing cuts.

“I would hope that the governor would recognize that not it’s very unfair to have municipalities pick up the additional cost of revenue sharing and then force them to go back out to spend tens of thousands of dollars more to initiate a referendum process,” he said.

But Maine’s Department of Education opposes the measure, meaning it may be at risk for a veto if it makes it to LePage’s desk.

Deputy Commissioner Jim Rier said he believes voters should have a chance to weigh in on how the new money is spent, whether it’s for increasing school programs or reducing taxes. About $1.9 million in new funding “seems like something that ought to be worth voter participation,” he said. Districts can also save the money and work it into their budgets next school year, he said.

Bangor City Manager Cathy Conlow said an additional referendum is unnecessary because the revised budget would reduce the taxpayer burden, something voters would obviously support.

The Bangor School Department budget approved last month would mean a tax increase of 43 cents per $1,000 of property valuation. With about $600,000 more from the state than originally proposed, property taxes would increase at a lower rate of 30 cents per $1,000 of valuation, she said.

“I think (voters) would be in favor of that instead of taking money and putting it into reserves,” she said. An additional vote would cost the city about $8,000, she said.

The Legislature will take up a slew of other bills Tuesday before they adjourn, including a technical correction to the state budget. Lawmakers also will attempt to override final bills LePage has vetoed, like one that would ban companies from determining health insurance rates based on where someone lives.