Gibson has been a mainstay in the Middlesbrough team in recent years, and is a graduate of the club's academy.
While his side were in the Premier League in 2016-17, he impressed many and was called up to the England squad in March 2017.
However, with his side in the Championship it is unlikely that he would get the chance in Gareth Southgate's team, so a return to the top flight could be a tempting offer.
Burnley's initial offer of £12 million was rejected, but 'Sky Sports' claim that the Lancashire team have upped their bid to £13 million.
Tony Pulis will definitely not want to lose one of his main players, but also admitted that it could be difficult to keep the club's captain at the Riverside.
"Everybody in their profession wants to play at the top level. He has to seriously think about it and that's him as a person, not Middlesbrough." the former Stoke manager said.
"What has got to happen is that we make sure we get the best deal we possibly can for the football club and that we know where the next centre half is."
With less than a week left in the transfer window in England, finding a replacement could be a difficult task.