The Turkey international, 28, confirmed earlier this summer that he would not be extending his stay with the 2013-14 Liga champions and had been linked with a number of Europe's top clubs.
However, Barca emerged as clear favourites for his signature in recent days and have now announced his arrival for an initial fee of €34m, with €7m to come by way of add-ons.
However, Barca retain the option of selling the player back to Atletico for a €4.1m loss before July 20 should the new president - set to be elected 48 hours earlier - not wish to sanction his arrival.
If Arda is kept on by the incoming Barca chief then he will not appear for his new club until 2016 due to the Fifa-imposed transfer ban.
The European champions are prohibited from fielding new signings until January after being found guilty of irregularities in the transfer of youngsters from overseas last year.
Arda has already been linked with a possible loan switch to Turkish champions Galatasaray until the winter break of next season in order to allow him to sustain form and fitness during Barca's suspension.