The 33-year-old Alves, who won 23 titles in his spell at Barca, is a free agent.
He has not indicated where he plans to go but the player's agent and former wife, Dinorah Santa Ana da Silva, said last week that she had met representatives of Italian champions Juventus as well as several other European clubs.
“I’m leaving but I’ll be back, because I’m a Barcelona fan just like anyone else,” Alves said in a post on Instagram.
“I’m a privileged, hard-working and honourable footballer who’s been lucky enough to wear the shirt of the best club in the world in a wonderful decade, thanks to the quality of its players and coaches.”
Barca's sporting director Robert Fernandez had announced Alves' departure at a news conference earlier on Thursday, saying: "This is a personal decision that the club has to accept."
He said the defender extended his contract for an extra season last year to ensure Barca had cover after they were banned from signing new players by world governing body FIFA for breaching rules on signing non-Spanish under-18 players.
But Fernandez said that even at that stage the question of Alves leaving this year was discussed.
"All we can do is respect his decision," he said. "For players that come here, this is their goal. We are not a club that buys players and sells them on. We are a destination."
Alves won three Champions League trophies, three European Super Cups, three Club World Cups and six La Liga titles with Barca. He also won domestic honours with previous club Sevilla.
As well as his thrusting displays down Barca's right flank and his on-field understanding with the talismanic Lionel Messi, Alves will be remembered for his exuberant personality.
"FC Barcelona express their gratitude to Dani Alves for the eight years that he has been at the club during which he has helped it to become bigger and better," read a Barca statement.
"His commitment to the club and his good nature has captured the hearts of Barca fans. For that reason, he will be invited to bid farewell to the fans at the start of next season.
"Now, FC Barcelona wish him all the best in this new phase of his footballing career."
(Additional reporting by Neville Dalton; editing by Ken Ferris)
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