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The 24-year-old Argentine went down when challenged by Erik Pieters to win a penalty for the Hammers.
West Ham captain Mark Noble converted from the spot to give the visitors a 1-0 lead.
The decision infuriated Stoke manager Mark Hughes, who is now under even more pressure following the 3-0 home defeat.
Lanzini has until 6pm on Tuesday to respond to the charge, which is officially known as the "successful deception of a match official" - in this case referee Graham Scott.
Hughes was understandably frustrated after the match at the penalty being awarded.
Jamie Redknapp’s Premier League team of the week: 16-17 December
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“It sounds like sour grapes when you talk about people getting punished retrospectively”
Stoke boss Mark Hughes
He said: "It's the way our luck is going at the moment.
"Things are conspiring against us and we need help from referees to get things right.
"I've seen it again. The guy's dived. He's drawn the challenge. He's a clever player. It was clearly a dive and the ref's seen something that no-one else saw. All in all, it was a poor sequence of events for him and for us.
"It sounds like sour grapes when you talk about people getting punished retrospectively. It doesn't help us now."
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If Lanzini is found guilty he will be handed a two-match ban ruling him out of the Carabao Cup quarter-final tomorrow night against Arsenal.
He would also miss the Premier League match at the weekend when David Moyes's side face Newcastle.