Gibson caught they eye last season, despite Middlesbrough's relegation, with the homegrown hero forming an integral part of a solid defensive unit.
The 24-year-old was an ever-present for the Tessiders last term and has attracted plenty of interest owing to his consistent performances.
The player received a call-up to the England squad at the end of last season and there have been suggestions that the player is tempted by a move that would allow him to play Premier League football ahead of next summer's World Cup.
Boro have already rejected offers worth over £20m for the defender this summer, with manager Garry Monk insisting that the club is under no pressure to sell players.
Monk said: "From day one, it was very clear we don't have to sell any player. I said all along it would have to be an extraordinary offer for any player to leave and we don't want our important players to leave."
Nevertheless, it is thought that a move to a club the size of City could be of interest to the life-long Boro fan, with Pep Guardiola having already admitted his interest in adding another defender to his ranks this summer.