Details of Leone's new club will be announced by them in due course, but media speculation has linked him with a move to Racing 92.
Glasgow Warriors head coach Gregor Townsend said: "Leone has attracted a lot of interest from other clubs due to the form he’s been in over the last year.
"During the Rugby World Cup a lot of clubs were interested in signing him and although we were looking to retain him, we have now agreed to let Leone leave a year early.
"Right from when there was interest in him I've been fully involved with Scottish Rugby in the discussions with Leone and his agent, and the deal which was subsequently agreed has allowed us to strengthen our squad for next season.
"Leone has been a fantastic player for the club and has been a pleasure to coach. He has worked hard and developed into one of the best second-rows in the world during the past three years.
"He has worked very hard to improve and has shown on numerous occasions how much he cares for his teammates and this club.
"We will rightly remember his outstanding game in last year's GUINNESS PRO12 Final, but his performance this season at home to Leinster was equally as impressive. Following an injury to Peter Murchie, Leone became our captain in the second-half and really stepped up for his teammates, making smart decisions and putting his body on the line in defence.
"We are looking forward to Leone delivering similar performances as we enter a very exciting run of games at the end of this season."
Speaking to glasgowwarriors.org, Leone said: "These types of opportunities only come around once and I couldn't turn it down.
"I'll miss all of the players, but I'll still keep in touch with them as I get on with them all so well.
"This is a new challenge which I'm looking forward to, but that is next season and my focus just now is trying to retain the GUINNESS PRO12 trophy and end my time at the club on a high.
"Glasgow Warriors have helped me a lot with the coaches and backroom staff improving my game and making me stronger.
"I'd also like to thank the Glasgow supporters for making me feel so welcome."