Former Reds defender Alan Hansen said the Uruguayan let down the club and manager Kenny Dalglish.
Dalglish, who has been fiercely protective of his player, said he did not see the handshake incident, which saw Manchester United defender Rio Ferdinand also withdraw his hand as Suarez approached him.
The incident comes after the FA recently took steps to "defuse further tensions" between QPR and Chelsea by
abandoning the handshake
prior to the FA Cup fourth-round tie between the clubs.
QPR defender Anton Ferdinand was expected to refuse a handshake with Chelsea captain John Terry, who has been charged with using racist language towards Ferdinand during Chelsea's 1-0 defeat in October.
Wayne Bridge also chose not to shake Terry's hand before Manchester City's match at Stamford Bridge in March 2010, after Terry was alleged to have had an affair with Bridge's girlfriend.
Jeremy Hunt, Culture, Media and Sport Secretary, described the episode at Old Trafford on Saturday as "incredibly depressing" and said it was up to the FA to scrutinise the matter further.
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