Since signing for the Ibrox club in the summer the 34-year-old midfielder has made public comments about the Parkhead club and its personnel, perhaps most notably when declaring that Celtic captain Scott Brown is “nowhere near his level”.
Thompson, who played for the Hoops between 2000 and 2007 and subsequently had a spell as coach at the club, advised his fellow Englishman to do his talking on the pitch.
Speaking at a William Hill media event in Glasgow, Thompson, who was sent off in his first Old Firm game at Ibrox, told broadcast media when asked if Barton could potentially be wound up at Celtic Park: “Joey could, yes.
“I know Joey well. I worked with him at Newcastle.
“He does do a lot of talking Joey, but if I can give him any advice I would say, ‘Go and do it on the pitch Joey, don’t do it off the pitch, do it on it’, because he is a good footballer.
“They (Celtic players) might try to capitalise on that with Joey and Rangers players might try to get at Scott Brown.
“But Browny has been there, seen it and done it.
“Joey is an experienced player, he has been around the block, but as I said before, don’t get involved in the hype, just play the game.”
Thompson was red carded in the 5-1 defeat by Rangers in November 2000, for picking up two bookings and he put it down to getting caught up in everything that surrounds an Old Firm game.
He said: “You just get involved in the hype during the week building up to it and you see the red mist. I wouldn’t recommend it.
“It wasn’t just my first game, it was a couple after that. You get caught up in the occasion, you get involved with one or two individuals.
“You mistime a silly tackle and it costs you and costs the team, costs the club and costs you points.
“Just play the game and don’t get involved in the occasion and all the hype around it.
“It’s fairly hypocritical of me to say that, but get on with it and enjoy the occasion.
“Don’t ruin it by sitting in the dressing room for part of the game.”