The chief backer of a Boston bid for the 2024 Summer Olympics is now supporting the idea of a 2016 referendum on the bid.

Suffolk Construction CEO John Fish, who is spearheading the privately funded nonprofit seeking to bring the summer Olympics to Boston, told the Greater Boston Chamber of Commerce on Tuesday morning that Boston 2024 is "committed to the highest level of transparency and accountability" and will gather signatures for a 2016 ballot question.

"Prior to this vote we will be working with the people of Boston and Massachusetts to build the best bid possible - one that reflects the best of our state and the Olympic and Paralympic movement," Fish said in a statement after the speech. "Then, the people of Massachusetts can make the final decision on whether we have achieved those goals."

The next statewide election will be held in 2016. Olympics supporters have watched their poll numbers plummet over the last three months.

Gov. Charlie Baker, a supporter of the ballot initiative process, said "having the 2024 people say they would support a statewide referendum, I think, is a good thing."

"And I think it's a signal and a message that they get the fact that the people of Massachusetts should have an opportunity to sign-on on this," Baker told reporters.

Asked whether he would get involved in the ballot campaign, Baker said, "The short answer to that is I'm not sure. It would depend to some extent on a lot of the other data that's being developed and the product that's being developed."

Baker and legislative leaders on Monday announced that they plan to hire an outside consultant by early May to evaluate the potential bid, at a cost of no more than $250,000.

No Boston Olympics, a group set up in opposition to bringing the games to the city, said in a statement that they were "glad to see Boston2024 embrace that idea after months of rejecting it."

"The ballot language itself now becomes incredibly important," the group said. "We hope to work constructively with Boston2024 to craft language that accurately and fully reflects the difficult choice facing our Commonwealth. We need to ask voters if taxpayers should be on the hook if things don't go according to Boston2024's plan."

In a statement released on Tuesday, shortly after Fish committed to a ballot effort, Boston Mayor Martin Walsh signaled his own reversal.

"The success of our bid for the Olympics depends on the support of residents and we should only move forward in a way that will bring the greatest benefit to the City and its neighborhoods. Over the next year, I encourage residents to engage in a conversation to learn more about what the Olympics could mean for Boston and the entire Commonwealth, and to put forward any suggestions or concerns," Walsh said.

Walsh's communications director Laura Oggeri later clarified that a referendum is Walsh's preferred way to measure public support for the games.

Walsh previously opposed a ballot question on the games, saying in January that there would not be a referendum and that organizers would talk to Bostonians, not just "ram it down people's throats."

Walsh told Boston Public Radio in February that he believes the Olympics would need 70 percent support and said 51 percent support would be inadequate.

Evan Falchuk, chairman of the United Independent Party and a 2014 gubernatorial candidate, has said he is aiming to mount his own effort to place a question on the Olympics before Bay State voters.

Using the 1984 Olympics as precedent, House Transportation Committee Chairman William Straus believes a voter referendum supporting cost controls on Boston's bid for the games would improve the city's bargaining position.

"What I'm hearing is not so much do you want the Olympics or not, but a real skepticism that these events will end up costing them money," the Mattapoisett Democrat told the News Service.

In 1978, nearly three quarters of Los Angeles voters approved a ballot measure prohibiting the spending of city money on the Olympics without guaranteed reimbursement, according to the LA84 Foundation, which was established with the surplus funds from the games.

Straus said the vote encouraged the private sponsors of the games to control spending, and he said a similar referendum could allay some concerns of those skeptical about bringing the Olympics to Boston in 2024. He suggested Olympics backers should support such a cost-control referendum.

"The whole tenor of the discussions would change if they were to be the ones passing that kind of ironclad guarantee for the people of Massachusetts," said Straus, who said there is still enough time for a similar move.

The first bid submission to the International Olympic Committee is set for Dec. 2016, according to Boston 2024.

The International Olympic Committee is slated to select a host city in summer 2017.