Gibson, who has made 185 appearances for Boro, signed a five-year deal with the club last season. But, per Sky Sports, the Clarets are looking to prise him away before the close of the current transfer window.
The 25-year-old has been at Middlesbrough since 2005 and emerged as an academy graduate five years ago. He has represented England from Under-17 level through to Under-21 and was called up to the senior side as a replacement for an injured Chris Smalling by Gareth Southgate for a World Cup qualifying match against Lithuania last March.
Sean Dyche's side aren't the only ones interested in Gibson, with Everton, West Ham United and Leicester City having contacted Boro over his availability this summer. But Burnley are supposedly looking to make the centre back their first signing of the summer.
Burnley and Tottenham are the only Premier League sides who haven't made a single signing since last season ended but are both understood to be looking to begin their transfer business soon. The Turf Moor side are reportedly keen on Gibson, while Aston Villa midfielder Jack Grealish is expected to join Mauricio Pochettino's side.
Dyche watched his side have their best-ever campaign in the Premier League era last season and is hoping to see them replicate the performances that had them finish in seventh place on the league table.
The Clarets conceded 39 goals over the course of the previous campaign, just three more than Spurs and one more than Chelsea. Bringing Gibson in would likely strengthen their already solid back line.