Grella joined the Shakers on a short-term deal at the end of last season following his release by Brentford and scored four times in 10 appearances.
Bury boss Richie Barker said: "Unfortunately we couldn't match some of the offers that Mike has received.
"It was always going to be a gamble from both parties on such a short-term contract, but throughout April I am sure his form alerted a number of clubs.
"Mike played a huge part in us retaining League One status and we can only thank him for that. He became extremely popular with the fans, the players and my staff.
"We all wish him the best of luck in the future and hope he has a successful career."