The PGMOL official is in hot water for comments he made about Liverpool and Jurgen Klopp.
There have also been videos shared of him appearing to sniff illicit drugs in a hotel room.
A PGMOL statement said: "We are following an internal process and taking the allegations into Coote’s conduct seriously as part of our ongoing investigation.
"Whilst David remains suspended, his welfare continues to be important to us and he is aware of the support network available to him."
Martin stated: "We're dealing with human beings and they will have some level of feeling or bias, conscious or unconscious, towards certain people and clubs and players because the way they're treated by them, I would have thought.
"I got on with certain referees way better than I did with others. There were one or two that I just couldn't build a relationship with as a captain even if I tried. What they do outside of their football career as a referee really has no impact in how much I trust them.
"I think everyone's going to have their opinions and their feelings. Unfortunately for David, he's been caught on camera expressing his, which is probably his biggest mistake. I can't judge anything else. I don't know what's gone on. I haven't seen it.
"I'm sure he's getting really criticised and I'm sure he's taking a lot of punishment by himself and everyone else for making a mistake. But everyone is a human being. Has it eroded my trust in referees? No, I think they're humans," continued Martin.
"They all make mistakes the same way we do and I do. My trust was probably eroded a bit more after the decision at Wolves than it did in watching a human being, a man make a mistake on a camera, so I think let's have a little bit of compassion and understanding.
"Everyone makes mistakes. He's done it outside of work. I'm sure he's going to pay the price for that in whatever way that is and I'm sure he's feeling very upset and embarrassed about it. I feel for the guy and what he's going through.
"I'm sure everyone will be enjoying the chance to criticise and make a statement about him. But I'm not going to. He's a human being. They all are. It hasn't changed how I feel about referees. Some of them I really like.
"Some of them don't like me. We're all different. We're all different so it is what it is and I'm sorry he's going through that and I'm sure he's very sorry as well."
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