The 27-year-old, who signed for the Bears in the summer of 2021, made 14 appearances for the club, scoring 42 points.
Director of Rugby, Pat Lam, said: “Tom has been a very popular member of the squad and we’re grateful for his contribution to Bristol Bears on and off the field during his time with us.
“With Tom already having agreed to join up with Tigers from the start of next season, and the opportunity arising there for a scrum half following Six Nations call-ups, the request came in to take him with immediate effect and we wanted to give Tom the opportunity to make the move early.
“With Harry Randall back from injury, Will Porter returning very soon and Andy Uren and Oscar Lennon in our ranks, we’re pleased to be able to give Tom the opportunity for some game time in the Premiership.
“We wish him all the best for the next chapter at Mattioli Woods Welford Road.”
Whiteley said: “I have really enjoyed the past 18 months at Bristol and feel I have grown as a player and a person. Bears is that kind of club, with that kind of culture.
“I would like to thank all my teammates, and Pat and all the staff, for everything they have done for me. I am also very grateful to the club for allowing me to grasp this opportunity at Leicester.”
