The 73-year-old businessman told an audience on Friday in New Hampshire - site of the first primary next year - that the country is in "grave peril" and that he "cannot sit quietly on the sidelines any longer".
"We have a president whose priorities are skewed toward promotion of himself rather than toward the good of the country," he said, according to the Boston Herald.
Mr Weld, who was the Libertarian Party's nominee for vice-president in the 2016 election, becomes the first Republican to announce a possible challenge to Mr Trump for the Republican nomination.
An official run could spell trouble for Mr Trump, even if he was to defeat Mr Weld in a primary contest. The last three presidents who have faced one – Gerald Ford in '76, Jimmy Carter in '80 and George W Bush in '92 – went on to lose the general election.