Webb, who has 28 caps for Wales and won two British and Irish Lions caps on the 2017 tour to New Zealand, has made over 140 appearances for the Ospreys since his senior debut in 2007.
His contract with the Pro14 side and the Welsh Rugby Union (WRU) ends in 2018 and Webb said it was a "huge wrench" to be leaving after spending his entire playing career with the team.
"I have made lifelong friends on and off the pitch and I cannot speak highly enough of the quality of the coaching team, players and all of the staff involved within the Ospreys organisation," Webb said in a statement.
"I will continue to give the region everything I can to make sure we have the best season possible and I would like to conclude by saying a heartfelt thanks to everyone within the Region for helping me achieve my goals within rugby.
"I would like to thank the WRU and the Ospreys for their efforts during this process."
The 28-year-old's move to France could hinder his international career as WRU's policy allows for the selection of only four players, that play for clubs outside Wales, to be included in the national team over the next two years.
Webb, who made his Wales debut in 2012, joins a long list of players that have recently left the region including Dan Biggar, Liam Williams, Taulupe Faletau, George North and Jamie Roberts.
(Reporting by Aditi Prakash and Shrivathsa Sridhar in Bengaluru; editing by Sudipto Ganguly)
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