The striker scored an injury-time equaliser for the Reds to maintain his record as the Premier League's top goalscorer but was earlier caught by TV cameras sinking his teeth into the Serbian's arm, for which he and the club have swiftly apologised.
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The 55-year-old told BBC Sport: "I don't think he will win that [PFA award] anyway because a lot of people can't differentiate between Suarez the player and Suarez the man.
"He's a world-class player but he gives you world-class trouble."
Suarez was involved in a similar incident for his previous club, Ajax, in 2010, when he was given a seven-match ban for biting then-PSV player Otman Bakkal and Lawrenson suspects that the striker could face a lengthy ban similar to the one handed out by the Dutch FA.
"If the Football Association decide[s] to take retrospective action and it's clear-cut, then Suarez will be in big trouble," he concluded.