ST. PAUL — The chief House sponsor of a bill to require a photo ID for voting in Minnesota said Monday she expects the Legislature to pass it soon — and that supporters are likely to bring the issue directly to voters if Gov. Mark Dayton vetoes it.

“That is absolutely still an option,” said Rep. Mary Kiffmeyer, R-Big Lake, the bill’s chief sponsor and Minnesota’s former secretary of state. Dayton, while not directly threatening to veto, has raised concerns about the bill, which critics have openly feared could make voting more difficult for senior citizens, college students and other young people who change addresses frequently, and new citizens.

Kiffmeyer said she still hopes to find common ground with Dayton on the issue. But, citing polls that have shown large majorities of Minnesota in support of such a requirement, she said a Dayton veto could result in sponsors re-introducing the bill as a constitutional amendment and sending it to the statewide ballot in 2012.

“We’re going to try to make it acceptable to the governor and to work with him,” Kiffmeyer said. “But there’s a strong base of support for this regardless of where he ends up.”

The House Transportation Committee approved Kiffmeyer’s bill Monday, one of numerous committee stops for the House and Senate versions of the bill. Kiffmeyer said floor votes by the full House and Senate could come as early as next week, after lawmakers return to the Capitol from a weeklong break that began Monday afternoon.

Supporters of the bill say the photo ID requirement is needed to make voting more efficient, cut down on voter fraud and bolster public confidence in elections. Under the legislation, the state would purchase electronic ID card readers for local governments and provide free photo IDs to those who can’t afford them.

Bill sponsors have estimated it would cost about $2.7 million to implement, although critics contend it could be more expensive. Kiffmeyer is hoping the Minnesota Secretary of State’s office could at least in part use federal funds from the recent “Help America Vote Act” to cover those costs, though officials from that office said they would prefer to use that money on established purposes including maintenance of the statewide voter registration system.

“We’ve heard from Congress there will not be additional appropriations under this act, so this will be the end of that money,” said Beth Fraser, government relations director for the secretary of state’s office.