A six-week absence would keep Gomez out until mid-January, with Liverpool facing nine games between now and then in three different competitions.
Dyche was responding to comments from Liverpool manager Jurgen Klopp regarding Burnley's tactics and the performance of referee Stuart Attwell.
Dyche sent his best wishes to the 21-year-old but insisted Mee had done nothing wrong.
Speaking in the video on Liverpool's Twitter account, Gomez said: 'Hi everyone, it's Joe, I just wanted to say thank you to you all for your kind messages and well wishes.
"Sean Dyche is rightly upset and annoyed by the comments. He would probably say it's possible but I think it's responsible that we have a little eye on that".
Klopp crticised the match referee Stuart Atwell, saying: "'I said to the referee after the first sliding tackle (made) over six or seven yards, when he got the ball and everyone liked it: "'Please, tell them they can not do that'".
"There are a lot of players who can play that position, especially in the way we play now".
But the German hit back in his press conference today: "Daniel Sturridge is no cheat".
Dyche also made clear that Klopp appeared to be in a minority of one in thinking there was an issue over the struggling home side's approach to defending against a team sitting second in the table.
"The media said we went about it the right way, tried to take on the game, tried to win a football match".
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