Ferguson branded Beck's performance "shocking" after he failed to flag for what the Scot thought was a "clear penalty" for the visitors when Wayne Rooney went down in the box after a challenge by Steven Caulker in Sunday's 1-1 draw at Tottenham.
The United boss said his club had "history" with Beck following his failure to rule out an offside Didier Drogba goal in 2010 and also claimed that: "There was no way we were going to get a decision from (Beck)."
The FA confirmed on Monday evening that they have contacted Ferguson asking him to clarify his comments. He has until 6pm on Thursday to respond.
Should the FA rule that Ferguson's comments amount to a questioning of Beck's integrity, they will charge the United boss with misconduct and he could be hit with a touchline ban and or fine.