Godin, 33, has played 387 times for Atletico since joining from Villarreal in 2010, scoring 27 goals - including famously clinching the club's first La Liga title in 18 years with a header in a 1-1 draw at Barcelona in 2014.
Godin did not say where he is going, but numerous Spanish media have reported that he is on his way to Italian Serie A club Inter Milan.
He is the club's longest-serving player, and its latest high profile servant to leave after Fernando Torres and former captain Gabi Fernandez departed last year.
Godin's departure will leave another hole in Atletico's defence, with France international Lucas Hernandez already agreeing to join Bayern Munich in June after they paid his 80 million euro release clause.
"I'm a little nervous, much more than I am before a game, but I have called everyone here to inform you that these are my final days at the club," Godin said.
"It feels painful to say goodbye, but I leave with my head held high after giving everything every day."
Atletico president Enrique Cerezo added: "We are here to say goodbye to an Atletico legend. You have been a vital part of this great era for the club, with your efforts and your goals, like the one which gave us the title at the Nou Camp.
"You have fought in a thousand battles and defended the club's shirt in every game and you have worn the captain's armband. Wherever you end up, Atletico wishes you a lot of luck. Atletico is your family, the Wanda Metropolitano is your home."
In addition to the La Liga triumph, Godin won two Europa League titles, one Copa del Rey, one Spanish Super Cup and three UEFA Super Cups with Atletico as well as reaching the 2014 and 2016 Champions League finals, only to lose both times to Real.
Godin also said he had turned down offers from other clubs to leave Atletico last year.
"I received offers, of course I did, but I decided to stay and I have no regrets about that. I wanted to stay at this club but we couldn't reach an agreement. I would love to play here until I'm 40 years old, but it's impossible," he added.
(Reporting by Richard Martin; Editing by Hugh Lawson)
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