AS says City are one of four clubs prepared to offer €120 million for the Argentina international.
City, along with fellow big spenders Paris Saint Germain, Anzhi Makhachkala and Monaco, are reportedly monitoring the situation and could move for him in the next transfer window.
The reports surfaced after it emerged last week that Messi had been unimpressed by the lack of support coming from Barca as he faces an alleged tax fraud case.
The 25-year-old and his father Jorge Horacio will appear for questioning in September as part of an investigation into alleged tax fraud in relation to the player's image rights.
"I am very calm," Messi said recently. "We have consultants who are handling these matters and we trust them. We hope to find a solution.
"I don't understand any of it and that's why we have lawyers."