City were quick to knock back Barca's initial approach. High-ranking executives at the English club have made it clear that the youngster is not for sale.
Pep Guardiola is particularly fond of Garcia and, while City hope to sign a new centre-back this summer, they also want to keep him involved as part of the first-team squad.
However, there is also an acceptance at the club that the uncertainty over Guardiola's long-term future could become a problem at some point. Players are also in the dark and with just one year left on his contract, Garcia is in a strong negotiating position.
Garcia, 19, was considered one of the brightest prospects in Barca's La Masia academy when he decided to leave in 2017 to join Premier League champions City, where he has a deal until 2021.
Guardiola handed him his full debut in a League Cup game against Leicester in 2018 and he has made 11 appearances this season, starting twice in the league and once in the Champions League.
He's also been capped five times by Spain under-21s and Barca have followed his development closely since he left three years ago.
A number of different sources at Barca said the club view Garcia as a "good fit" centre-back for the team's style of football. He was brought up on the club's philosophy and one of his major strengths is his ability on the ball.
Barca, who missed out on Matthijs de Ligt to Juventus last summer, have been searching for a central defender for a long time as they contemplate a future without Gerard Pique, who turned 33 in February.
City's reluctance to sell makes a move for Garcia this summer unlikely, especially as Barca expect to be less active in the transfer market due to the effects the coronavirus pandemic has had on their budget. They will focus on their main target, Inter Milan striker Lautaro Martinez, and will not have money left over to invest elsewhere.
But sources told ESPN that that Barca will not give up on Garcia in the long-term and will continue to monitor his situation as he moves into the final 12 months of his contract at the Etihad Stadium.