The 46-year-old Russian coach was sacked by the Sky Bet Championship club earlier this week as a result of a poor run of form.
Chelsea are looking for a new technical director after Michael Emenalo ended the best part of a decade working at Stamford Bridge by leaving the club last month.
Chelsea are undecided whether the role will stay the same or be split into two separate roles.
The 46-year-old was axed by Championship side Hull on Sunday after only six months in charge of the team, with the club struggling to meet lofty expectations - now sitting 20th in their domestic table.
Slutsky, who regularly watched Chelsea games last season, signed Blues trio Ola Aina, Michael Hector and Fikayo Tomori on loan at Hull.
The Russian boss only took control of the side in the summer, but after just four wins in 20 games, he departed the KCOM Stadium on Sunday.
Slutsky said: "I would like to really thank everyone associated with Hull City from the players and staff to the owner and especially the fans".
"I am very proud to have been through this experience, it was challenging but a very exciting time for me".
He has left the club along with the Tigers' head of club strategy Oleg Yarovinsky.