Garcia, 19, is a priority target in Barcelona’s plan to regenerate their squad following a disappointing season.
The former La Masia player moved to City in 2017 and became a regular in Pep Guardiola’s side last season.
His good form for the Citizens has led to Luis Enrique handing him his first call-up for the Spain national team for upcoming games against Germany and Ukraine.
Mundo Deportivo claims Barca don’t want to pay a big fee for Garcia because his City contract ends next June.
Barca willing to wait
The Barcelona-born defender has told Guardiola that he does not plan to renew his contract, pressuring the English club.
Even so, their €10m offer is well short of City’s asking price of €33.3m (£30m).
Barcelona are now awaiting City’s response to their offer and if it is rejected, they will wait for the final days of the transfer window.
If City still refuse at that point, Barca will wait for Garcia to arrive on a free transfer next summer or for a small fee in January.
Garcia made 20 appearances for City last season, including a start in their Champions League quarter-final defeat to Lyon.