Suarez will miss Liverpool's Merseyside derby with Everton on Sunday due to his 10-game ban for biting Chelsea defender Branislav Ivanovic.
Former Liverpool midfielder Thompson, who also had spells with the likes of Coventry, Blackburn, Wigan, Portsmouth and Bolton, had anger issues of his own and says the desire to win can often cloud your judgement.
"In my younger years I just went with it and let it go. I did not really understand it," said Thompson on Goals on Sunday.
"As you get older you learn to keep your calm a little bit more. I can understand some of the stuff that Luis Suarez does - the way he plays the game; he is just a born winner and wants to do everything he can to win that game.
"When things are not going right you just get really angry about it and you don't know how to channel that frustration. But it is about trying to keep calm and focused and remember what you are there for. Looking at triggers - its derby day today and that always got me over-anxious and a bit het-up.
"It's about keeping control because one moment of madness and you can let the whole team down."
Thompson was on the field when Steven Gerrard announced himself to the world by scoring his debut goal in Liverpool's 4-1 win over Sheffield Wednesday in 1999, and says he has never played with a better player.
"The Premier League was taylor-made for Stevie G, he was big, he was powerful and he was leggy. He was a special player and he had a real drive and determination. Some people thought that it was borderline arrogance but it wasn't it was a real inner-confidence. You just knew that when he was going out on the pitch that something was going to happen."
Thompson left Liverpool in 2000 to move to Coventry and despite the club getting relegated, he said it was a good move for his development.
"The season after I left Liverpool they won five trophies and I got relegated - that was easy to take when I was watching that," joked Thompson.
"But in terms of development I felt I made the right decision because when I was at Liverpool I was probably playing out of position on the right hand side. When I left, it allowed me to develop my game a little bit more, to play in different positions. It brought me on and the season after it led me to get an England call-up. So in terms of my development I made the right decision but I did miss Liverpool and I did regret it."
Thompson's career came to a premature end due to injury and he has some advice for young players who think their football careers will never end.
"It is difficult to take - I did manage to keep on going for another four or five years but they were not great years. I did not let anyone down but psychologically it does have an impact because you cannot be the best you can be - you know you can be a lot better but physically you cannot do it.
"I would say players need to look after yourself. I think in my early days my diet was not correct, I think I probably drank a little bit too much alcohol - not to an excess, but at the end of the day you are an elite athlete and you should banish it from your diet."