Gibson, 25, made 51 appearances in all competitions last season, as Boro reached the Championship play-offs.
The season before, Gibson was one of Middlesbrough's stand-out performers despite their relegation from the Premier League. Gibson even received an England call-up in March 2017, but didn't make an appearance on the pitch.
Leicester's interest in the English defender comes due to the possible departure of another English centre back. Maguire was a rock at the back for an England side that made the World Cup semi finals, and in doing so has caught the attention of Jose Mourinho's Manchester United.
The Mail report that Leicester are not willing to let the Sheffield born centre back depart the club, with Claude Puel recently voicing his intent to keep him at the club as quoted by Four Four Two, however Maguire is reportedly in favour of the move to Old Trafford.
United are willing to pay a whopping £65m for Maguire's services, so it remains to be seen whether Leicester will be enticed by such a tempting offer.
Despite the transfer windows imminent closure, Leicester still have a lot of business planned, with Leonardo Ulloa, Islam Slimani, Ahmed Musa and Nampalys Mendy all set to depart the King Power Stadium.
Should Gibson sign for Leicester, Foxes fans will be hoping that he acts as a partner to Maguire, not his replacement.