Piara Powar, executive director of European football's anti-discrimination body FARE, claims some supporters are reacting in a worrying manner after Liverpool mounted a sustained defence of their Uruguayan striker.
Suarez was found guilty of racially abusing Manchester United defender Patrice Evra by a Football Association-appointed independent regulatory commission last month, and is currently serving an eight-match ban.
Powar has previously criticised Liverpool for their response to the Suarez affair, and earlier this week he urged the FA to charge Liverpool and manager Kenny Dalglish with bringing the game into disrepute.
Powar told Sportsweek on BBC Radio Five Live: "If you look at the comments Dalglish has been making, if you look at how Liverpool resort to defend what they see as an injustice, look at the way their fans have been whipped up into a tribal fervour.
"Some of the words being used to describe the FA processes, some of the way to describe the FA and its role in governance on these sort of issues, that is really beyond the pale."
Powar claims Liverpool are not setting an appropriate example to their supporters, adding: "The environment at Anfield is so driven - in terms of the paranoia that some fans feel - [by] expressions of injustice from Liverpool, especially over the last few weeks."
He added: "There is a wider issue here of the way Liverpool have dealt with this and it has disconcerted many observers.
"It is a constant undermining of the FA's role as the governing body in football."
Police have investigated allegations that Oldham defender Tom Adeyemi was racially abused in Friday's FA Cup match at Liverpool.
A 20-year-old man from Aintree was arrested on suspicion of a racially aggravated public order offence following the alleged incident.
Liverpool, who beat Oldham 5-1 on Friday, last night said they were also investigating the incident involving Adeyemi.
The club issued a statement saying they are continuing to work closely with the police to establish the facts surrounding the incident, saying they have supplied evidence to police, including CCTV footage and still photography, plus statements from stewards.
Powar is also concerned by the levels of abuse aimed at footballers on social networking websites.
Powar does not think the problem of racism is getting worse but said the problem is generated partly by "the way in which fans feel they can get on something like Twitter and make comments without accountability".
He added: "There is nobody touching them on their shoulder looking at their comments and saying, 'No, that is not what is acceptable'.
"I don't think the situation has been worse than it's been for a long time but we are dealing with difficult issues here between players and the way in which the clubs have responded."