The 'Clarets' had previously seen two bids worth £12m and £13m turned down in the last 48 hours, but they appear to have reached an agreement at the third time of asking.
Gibson came through the academy at the 'Boro' and made 51 appearances last season as they fell agonisingly short in their quest to return to the Premier League at the first time of asking.
Despite a season in the second tier, Gibson's stock remains high, mainly thanks to a strong maiden Premier League campaign in 2016/17, when his performances earned him a call-up to the England senior squad.
The 25-year-old will now travel to Turf Moor for talks as Burnley close in on their first transfer breakthrough of the summer.
The fee will provide Tony Pulis' side with vital funds as he looks to build a squad capable of staying the course in the race for promotion.