Antonio Conte targeted Bakayoko as their first choice option to strengthen their midfield and they are now expected to allow Nemanja Matic to leave to make room for the France international.
Talks between Chelsea, Monaco and Bakayoko's representatives have moved slowly since they first started but all parties ultimatley came to an agreement for the 22-year-old.
Bakayoko returned from his holiday in New York to complete the final arrangements of the transfer, though a knee injury could rule him out of the Blues's pre-season friendlies.
The Frenchman, who will wear the No.14 shirt, told Chelsea's official website: "I am very happy to be here and to join this great team. I have grown up watching Chelsea. Signing was the natural thing for me because it is a club I loved very much in my childhood.
"I am now looking forward to evolving under this fantastic coach and alongside so many great team-mates."
He will become the club's second most-expensive signing of all time as he follows goalkeeper Willy Caballero and defender Antonio Rudiger into Stamford Bridge this summer.
His performances for his club have so far only earned him one appearance for his country as he competes with the likes of N'Golo Kante and Paul Pogba for a starting spot.
"Despite his young age he already has plenty of experience as a French title winner and having competed in the latter stages of the Champions League, we believe he will add considerable quality to our squad," Chelsea technical director Michael Emenalo said.
The Blues will next look to complete a deal for Juventus's Alex Sandro, which could break their transfer record that currently stands at £50m for Fernando Torres, as well as looking for attacking options.