The picture appeared on the Dutch forward's official page on social networking site Twitter just hours after Manchester United beat Liverpool 1-0 in the third round of the FA Cup -- a match refereed by Webb, who was in charge of last year's World Cup final.
Webb angered Liverpool by awarding home side United a first-minute penalty from which Ryan Giggs scored the only goal of the game at Old Trafford before sending-off Steven Gerrard, the Merseysiders' captain.
An FA spokesman said: "We will definitely look into this matter."
As well as the photo, Babel wrote: "And they call him one of the best referees? That's a joke. SMH (shaking my head)."
New Liverpool manager Kenny Dalglish labelled Webb's penalty decision a "joke" and disputed the red card shown to Gerrard.
But the Scotland great's son, Paul Dalglish, went even further.
Paul Dalglish, like Babel, took to Twitter to slam Webb, whom he suggested was unduly influenced by United manager Sir Alex Ferguson.
"Fergie has his puppet Howard Webb on a piece of string," said the former Newcastle striker, who also mocked the recent award to Webb of an MBE (Member of the Order of the British Empire) medal in Britain's New Year honours list.
"Howard Webb MBE. Manc of the Busby Empire," said Dalglish junior in a reference to legendary United manager Sir Matt Busby.
Babel's comments were the second time this weekend a current Liverpool player was involved in Twitter controversy.
Glen Johnson, who missed Sunday's match as his wife had gone into labour, attacked TV pundit Paul Merson after the former Arsenal and England midfielder criticised his performances for Liverpool.
"Comments from alcoholic drug abusers are not really gonna upset me and who is Paul Merson to judge players, he was average at the best of times," Johnson wrote on Saturday.
"The only reason he's on that show is coz he gambled all his money away. The clown!," he added.
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