Hull are allegedly keen to make a final decision as soon as possible and it has been claimed that an announcement could be made in the next 24 hours.
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.
The defending champions are yet to decide whether Emenalo, who is now at Monaco, should be directly replaced by Slutsky or whether the role should be shared between the two.
However, his time at the club was short lived after winning just four matches in the Championship, which has left the club in 20th place on the table.
Slutsky was granted access to Chelsea games last season and visited the club's Cobham training ground.
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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".
Hull now sit 20th in the Championship table, and are without a win in five games, recently drawing 2-2 with Sheffield Wednesday on Saturday.
The club confirmed Slutsky's departure in a statement and also said head of club strategy Oleg Yarovinsky had left his role at the KCOM Stadium.
"I am very proud to have been through this experience, it was challenging but a very exciting time for me".
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