Goal first revealed prior to the Premier League transfer window closing that the France international had opened talks with the Rossoneri.
Both sides of an agreement have now confirmed that a deal is complete, with Bakayoko set to spend the 2018-19 campaign in Italy before another decision is taken on his future.
“The 23 year-old joins the club on a loan deal with an option to make the move permanent.
“The midfielder has two major honours to his name, winning the league title in 2017 with Monaco and then the FA Cup with Chelsea the year after.
“Bakayoko will be unveiled to the media in a press conference on Friday August 17th.”
Chelsea only signed Bakayoko for £40m ($51m) in the summer of 2017.
He was, however, to find the going tough in England and began to see his form and ongoing presence in the side questioned.
Maurizio Sarri has deemed him surplus to requirements since inheriting the managerial reins from Antonio Conte at Stamford Bridge.
That has led Bakayoko to Milan, with the opportunity now there for him to restore confidence in his game alongside the likes of Gonzalo Higuain and Franck Kessie at San Siro.