The Scottish Daily Record says Messi's representatives held talks with City officials in a Barcelona restaurant last week over a potential transfer, with the Argentina international growing increasingly dissatisfied with the state of the Catalan club's squad.
Barcelona were rocked by the world-record move of Neymar to Paris Saint-Germain earlier this month and are yet to sign a replacement for the Brazilian, with overtures for Philippe Coutinho and Ousmane Dembele being rejected by Liverpool and Borussia Dortmund, respectively.
Messi verbally agreed to sign a new four-year contract with Barcelona before Neymar left for PSG, but is yet to put pen to paper on the agreement. His current contract expires next summer.
Barcelona sources told the Daily Record that the 30-year-old "is seriously considering leaving".
"He is open to a move, but knowing the player it will take him a while to decide," the source added.
City bosses have long been interested in bringing Messi to the Etihad Stadium, with manager Pep Guardiola also in favour of the move.
Guardiola said last year that he would be the first person to call the Argentina star if he indicated a willingness to leave the Nou Camp.
"In the case of Leo, I really wish him to play here and finish his career here [at Barcelona]," the former Barcelona coach was quoted as saying by the Daily Telegraph in October 2016.
"But maybe he will think, 'OK, I want to go play somewhere else,' because maybe he wants his kids to speak English or live a different experience, like has happened to many of us.
"Maybe it could happen, but if he decides to go and it happens, there would be a list of seven or eight clubs that would want to sign him, and it would be down to him. Messi will decide where to go."