Gibson linked up with Boro at the age of 12 and has since gone on to make 203 appearances for the North-East outfit in all competitions.
The 25-year-old also has just over four years remaining on his contract, but Boro's failure to earn an immediate return to the Premier League has left the club open to bids from interested clubs.
According to the Northern Echo, Southampton will become the first team to make an official approach for the player, with manager Mark Hughes known to be an admirer of the Englishman.
Hughes wanted to sign Gibson during his time at Stoke City, only for Middlesbrough to insist on Gibson remaining at the Riverside Stadium after relegation to the Championship.
As it stands, it is being suggested that Boro are yet to decide on how they will handle interest in the academy graduate.