The Clarets said that the fee for Gibson equals the club record transfer paid last year for striker Chris Wood and that Gibson had signed a four-year contract.
The 25-year-old has been with Middlesbrough since joining their youth system and was part of the team that was promoted to and then relegated from the Premier League in 2017.
Gibson, the nephew of Middlesbrough chairman Steve Gibson, will join Ben Mee and James Tarkowski as Burnley's main central defenders.
Burnley manager Sean Dyche has been keen to increase the size of his squad given the increased demands caused by participation in the Europa League.
Media reports suggest that Burnley will complete the signing of goalkeeper Joe Hart from Manchester City on Monday to give them three England international keepers.
Nick Pope, who was part of England's World Cup squad, has had an operation on a dislocated shoulder and is expected to be out for several months while Tom Heaton has just returned from his own shoulder injury and is out with a calf strain.
Hart, who fell out of favour at City following the appointment of Pep Guardiola, has spent the past two seasons on loan at Torino and West Ham United.
(Reporting by Simon Evans, editing by Pritha Sarkar)
