AUGUSTA — School districts across Maine will be watching Augusta this week as lawmakers consider a bill that would let districts revise their 2013-14 budgets to spend unexpected state aid – without opening the polls to seek voters’ approval.

The measure is among the final bills the Legislature will take up when it reconvenes Tuesday, along with bills that have been vetoed by Republican Gov. Paul LePage.

Critics of the school bill say voters should have a say in how the extra money is spent. Supporters say that holding extra school budget votes would be a financial burden for cities and towns.

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

Many school districts approved their budgets before the Legislature enacted the $6.3 billion, two-year state budget on June 26, so they weren’t sure how much funding they would get from the state.

With an extra $28 million going to schools this year, some districts will get over $1 million more than they expected. At the same time, the budgets of some districts — including Portland and Bangor — don’t account for additional teacher retirement costs that the state once paid.

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

Portland will receive $1.9 million more next school year than what was proposed in LePage’s budget. It also will have to pay $1.3 million in teacher retirement costs that weren’t budgeted. Brennan said that reopening the polls would cost Portland $12,000 to $15,000.

Maine’s Department of Education opposes the measure, so it may be at risk for a veto if it reaches LePage’s desk.

Deputy Education 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 could just 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 taxpayers’ burden, something voters would obviously support.

The Bangor School Department budget, approved last month, would increase property taxes by 43 cents per $1,000 property valuation. With about $600,000 more aid from the state than originally proposed, property taxes would increase just 30 cents, 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 other bills Tuesday before it adjourns, including bills that LePage has vetoed. A veto override takes two-thirds votes in the House and Senate.