Messi has been linked with a move away from the Camp Nou when his contract expires in June, with reports suggesting the Premier League club has already tabled an offer.
However, City haven't offered the Argentine superstar a contract and have no plans to open negotiations, a source told Stone.
Speculation over the 33-year-old's future has intensified with Barcelona's Champions League campaign on the brink of ending. Messi's side suffered a 4-1 loss in the first leg of its last-16 meeting with Paris Saint-Germain, who are also considered favorites to land the six-time Ballon d'Or winner.
Messi stunned the football world last summer when news surfaced that he wanted to leave Barcelona. City were considered the front-runners to sign him before he decided to stay for one more season to avoid a legal battle with the Catalan side.
Though City were heavily linked with Messi last summer, they didn't make an offer to sign him, Stone reports.
Barcelona boss Ronald Koeman said earlier this month he's "not confident" about Messi's future at the club. The Dutch manager revealed his doubt just days after Messi's €555-million contract was controversially leaked to the press.