The Germany international, who is out of contract this summer, now seems unlikely to renew terms with the treble winners.
City had offered the 32-year-old a new one-year contract with the option of a further 12 months but Barca are understood to have presented a more tempting three-year, and more financially-rewarding, package.
City are comfortable with the situation and the player would leave with their blessing.
The development comes on the same day City are understood to have agreed a £30million deal with Chelsea to sign Croatia midfielder Mateo Kovacic.
The champions are set to pay an initial £25million for the 29-year-old with a further £5m to follow in potential add-ons.
Kovacic, who joined Chelsea from Real Madrid in 2018, is entering the final year of his contract at Stamford Bridge.
City now expect to complete formalities with the player and an official announcement will follow in due course.
Kovacic’s versatility and experience would go some way to compensating for the loss of Gundogan.
The influential former Borussia Dortmund midfielder has won five Premier League titles, two FA Cups, four League Cups and the Champions League in his seven years at City.
This year Gundogan lifted three trophies as City became only the second English side to win the treble.
The club had been keen to retain his services but, given his age, did not want to match Barcelona’s offer and have taken a pragmatic decision.
Kovacic would arrive at City as a four-time Champions League winner having been a member of Chelsea’s victorious side in 2021 and triumphed on three occasions with Madrid. He also had a spell with Inter Milan earlier in his career.
More from Sporting Life
Fast Results
Scores
