The legal action, filed in a Florida federal court, also accuses Cohen of making "improper, self-serving, and malicious statements about his former client, his family members, and his business".
Cohen worked as Mr Trump's attorney for over a decade. He was also a vice-president at the Trump Organization, and was often described as Mr Trump's fixer.
But the two had a major falling out after the 2016 election, as investigators began looking into several of Mr Trump's aides.
In 2018, Cohen was sentenced to three years in prison and a fine after pleading guilty to charges of fraud and campaign finance violations.
Now out of prison, Cohen has become a high-profile critic of Mr Trump, and a frequent guest on news programmes.