Sarr, who arrives as a free agent, becomes manager Frank Lampard’s fourth signing of this summer.
The 21-year-old has signed a five-year deal with the Blues.
The French defender will, however, spend the 2020/2021 season on loan, likely at Bayer Leverkusen, as Chelsea try to secure Kai Havertz from the Bundesliga club.
Sarr scored three goals in 119 appearances for Nice.
His arrival comes a day after the £50 million capture of Ben Chilwell from Leicester City, with more signings expected in the coming weeks.