The Irishman joined the Hoops on loan in February and scored 16 goals in 19 appearances.
However, the club endured a disappointing season, finishing some way behind Rangers in the SPL.
"It's been a strange season for the club," he told Celtic's official internet magazine, The Huddle Online.
"But this is something I have always wanted to do and I'll never regret it.
"I have no regrets in football and I've loved my time here.
"From a personal point of view, I've scored a lot of goals and I'll look back with a lot of fond memories."Keane has also given his support to Neil Lennon as the race to become the next Hoops boss gathers pace.
He said: "I really hope that Lenny gets the job.
"He has all the attributes to be a great manager.
"He is very vocal and knows his football and one thing he has is a passion for this club. I personally hope he gets it," he added.