Laporte, 23, had a 65 million-euro (57 million pounds) clause in his contract with the Basque side, which he activated ahead of joining Pep Guardiola's City for whom he will become a club record signing.
Spanish and British media reported that Laporte will be presented as a City player on Tuesday as the Premier League leaders bolster their back line by recruiting the France Under-21 player.
"Athletic thanks the player for what he gave during the time he was at our club," Bilbao said in a statement.
At 57 million pounds, Laporte will overtake Kevin De Bruyne (55 million pounds) as the club's most expensive signing.
Frequent injuries to Manchester City captain Vincent Kompany and Guardiola's lack of faith in Eliaquim Mangala meant the Catalan coach wanted to add another central defender to challenge Nicolas Otamendi and John Stones.
Laporte has been a regular in the Athletic side since breaking through in the 2012-13 season, winning the 2015 Spanish Super Cup.
"I want to thank you for everything you have given me," said Laporte in a message attached to Athletic's statement.
"I arrived as a teenager and here I have grown into a person and a football player. I don't want my departure, ahead of a new challenge in my professional career, to be a goodbye but a see you soon. Here I leave a unique, different club, and one I will never forget."
(Reporting by Rik Sharma, editing by Ed Osmond and Christian Radnedge)
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