Gibson has played just six times for the Clarets since joining in 2018.
Watford were rejected with three offers to sign the centre-back.
"The defender made it clear, in the weeks leading up to the end of the January transfer window, that he wanted to leave Turf Moor to seek regular first team football.
"Unfortunately for the player, the club did not receive an offer it felt was acceptable and he remains contracted to the club.
"The club will not be making any further comment on this matter."