Gibson, the nephew of Boro owner Steve Gibson, is widely considered to be one of the best players outside of the top flight, having received an England call-up last season.
The 25-year-old came through the ranks at Boro and has been ever-present this season for his hometown club as they look to return to the Premier League at the first time of asking.
According to 'Sky Sports News', Everton, West Ham and Leicester are all keen on the centre-back, with the 'Foxes' having seen approaches knocked back in the past.
Despite suffering relegation from the Premier League last season, Gibson showed his loyalty to the club by penning a new five-year deal in the summer and it is believed that the Teessiders would be reluctant to sell "one of their own".
In order for them to do so, it would likely take a fee of upwards of £20m to convince Boro to cash in.