BOSTON — Stung by recent polling that showed only a third of Boston residents want the city to host the 2024 Summer Olympics, the group organizing the bid has reversed itself and now supports a voter referendum on the proposal, an inauspicious development for an already shaky bid.

John Fish, chairman of Boston 2024, the private group that organized the bid, announced Tuesday morning that the group would seek a statewide vote and even help gather the signatures required to get the measure on the ballot. Mr. Fish, who is also chief executive of the region’s leading construction firm, pledged that if the measure failed, or if it failed in the city of Boston, the group would drop the bid.

His stance is a reversal of the group’s earlier position that no referendum was necessary and that even if one were held, Boston 2024 would not necessarily abide by it if it were nonbinding.

In a statement Tuesday, Mr. Fish said the Games would be “good for Boston and will create thousands of jobs, drive economic development and serve as a catalyst for the long-term plans of Boston.” He said Boston 2024 was backing a vote because it was “committed to the highest level of transparency and accountability.”