Suarez was caught on camera biting Chelsea defender Branislav Ivanovic during Sunday's 2-2 draw at Anfield in an incident that went unsanctioned at the time, going on to score an equaliser deep into stoppage time.
The 26-year-old is expected to receive a lengthy ban from the Football Association and may well miss the rest of the season.
Although manager Brendan Rodgers was reluctant to comment immediately after the match, the striker quickly apologised both to Ivanovic and publicly and Liverpool condemned the actions of Suarez and fined him.
The Reds were heavily criticised for their handling of the situation when Suarez was found guilty of racially abusing Manchester United's Patrice Evra last season, with the club robustly defending their player.
Thompson, a former player and assistant manager at Anfield, is pleased Liverpool have reacted quickly and decisively this time.
He told Sky Sports News: "Luis apologised, then we had the club apologising and now they've fined him very quickly. In the FA's eyes, they will look at Liverpool Football Club as having acted in the right manner as befits our club.
"But in terms of thinking, 'We may get away with a lesser ban', I don't think anything like that's been done.
"Luis Suarez will get a massive ban, 10 games plus or maybe even more, maybe a monthly ban as Eric Cantona got. We have to be willing and ready to accept that."
Comments by managing director Ian Ayre today seem to draw a line under speculation Liverpool could sell Suarez this summer, and Thompson is happy to accept whatever the club decide.
He said: "I will go along with the football club. If the football club says it's the last time he plays in a red shirt at Liverpool, I will go along with that. If he stays, I will go along with that.
"The biggest thing is the player himself. He cannot allow anything like this to happen again."
Suarez was banned for seven matches in Holland in 2010 after another biting incident when he was an Ajax player, and former Liverpool defender Alan Hansen criticised the striker for shaming the club.
Speaking on Match of the Day 2, Hansen said: "You're not expecting anything like this. This is absolutely appalling. Why he did it, we will never know. He's got previous, there's obviously a massive flaw in the character.
"To watch these pictures that will be going right round the world is horrific for Liverpool Football Club, their history, their tradition, and for the player himself. When he sees these images, he'll be apoplectic. It's just appallingly bad."
The Professional Footballers' Association has offered Suarez anger management counselling, and former Liverpool striker David Fairclough believes the Uruguayan is worth persevering with to improve his behaviour.
He told Sky Sports News: "Clearly Luis needs a little bit of help, he has an anger issue I suspect, that's his drive to win and compete.
"I think Liverpool have to try and work with Luis Suarez, he's shown improvement in other ways over the last 18 months to two years and hopefully they can find a suitable solution."
Another former Liverpool player, John Barnes, thinks the reaction to Suarez's bite has been over the top and described suggestions he should not play for the club again as "ridiculous".
"He has to suffer the consequences of his actions - that's all he has to do," Barnes told BBC Radio Five Live.
"I've heard talk about him not playing for Liverpool again, which I think is ridiculous. I've heard talk about him bringing the good name of Liverpool down, which is also ridiculous - he's brought his own good name down."
Marseille midfielder Joey Barton, meanwhile, believes Liverpool should support Suarez because it is his dark side that makes him such a good player.
Liverpudlian Barton wrote on Twitter: "Suarez is a good as a player as there is in world football at the minute. Yes, he's messed up and shouldn't have bitten another player but..
"...a player like him has to play right on the edge. Without it he wouldn't be the talent he is. LFC would be mad not to back him IMO..."