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PSG have also been linked with the six time Ballon d'Or winner but Pep Guardiola's side have remained a constant potential destination, due to their finances and the bond between the manager and Messi.
Garcia on the other hand has been expected to leave Manchester and head in the opposite direction, with his contract up at the end of the season.
The centre back has not yet confirmed a deal with the La Liga side but he dropped his biggest hint yet about his future on Tuesday.
"Everyone knows that Messi is the best player in the world," he said at a press conference ahead of Spain's 2022 World Cup qualifiers this week.
"It would be special for anyone [to play alongside him], week after week what he does is incredible."
Asked if he had an agreement in place with new Barca president Joan Laporta, the 20 year-old added, "No, neither Laporta nor anyone else has contacted me.
"I made the decision not to renew [with City] - and what I will do, I don't know. The only option I have right now is to focus on the present."
Barca manager Ronald Koeman wanted to sign Garcia last summer and again in January, but the club didn't want to spend money on his transfer.
Instead the centre back can join this summer coming for nothing, with the club's financial difficulties suggesting that could be a theme for the Catalan side at the end of the season.
City's Sergio Aguero is another player that's been linked with the Spanish side, with the Argentina forward's contract also up at the end of the season.
Despite going out of this season's Champions League, Barcelona's form has improved in recent months and Messi continues to be at the heart of that.
The 33-year-old continues to break records and there has been talk of him staying at the Nou Camp beyond this summer, but he still needs to agree a new deal.