Former Massachusetts Gov. Mitt Romney told a group of donors on Friday that he is considering a White House bid in 2016, a move that could scramble the race for the Republican presidential nomination.

The remarks confirm months of speculation that the GOP’s nominee from 2012, who was also a top 2008 contender, is seriously weighing a third White House bid.

Mr. Romney’s comments to donors at a meeting in New York City came at a provocative moment—just as other Republicans are taking concrete steps to pursue the nomination. Many top GOP donors have started to rally behind former Florida Gov. Jeb Bush, setting up a potential clash between the party’s last standard-bearer and the most storied family in Republican politics.

Messrs. Romney and Bush, should they both run, would contend for many of the same donors who favor pro-business candidates from the party’s establishment wing.

Despite opening the door to another bid, Mr. Romney, 67 years old, is still considering his options, confidants say. At Friday’s meeting, he didn’t give a timetable for when he would make a final decision, but pointed to some reasons for another run, citing unrest overseas and concerns about the long-term health of the economy.