The 27-year-old was wanted on loan for the rest of the season by the Hornets.
However, multiple outlets - via the Watford Observer - in the UK are reporting the move is not sitting well with the Clarets.
They do not want to lose Gibson unless they can secure a replacement.
He only moved to the club from Middlesbrough 18 months ago, but has only played one time for the team this season.
Gibson would like to move so that he can resurrect his career, but Sean Dyche may want to hold onto him for the rest of the season.