The Premier League official is in hot water as a video from a few years ago has surfaced.
In the video, Coote is talking to a friend and discussing both Liverpool and their ex-boss Jurgen Klopp in a derogatory way.
On Coote, who is facing a disciplinary hearing, Walton told the Times: "There are some Premier League referees I consider more loose-lipped, and more laddish in their behavior away from the public spotlight.
“David is not one of them. I would have described him as captain sensible, maybe a bit highly-strung at times, and not one of the lads.
"Unlike some referees, you rarely see him laughing and joking with players during matches — he is generally very straight-laced. I look at the video and wonder if part of the problem is that he is trying to be someone he is not."