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 yesterday'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 yesterday that: "There was no way we were going to get a decision from (Beck)."
The FA confirmed this 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.
Rooney was convinced he should have been awarded a penalty, and his team-mate Rio Ferdinand agreed.
"I cannot believe he didn't give it. The linesman (Beck) didn't really help us on the near touchline at all in the second half especially.
"He maybe wasn't going to put his flag up but I am disappointed that the referee didn't give what was a stonewall penalty."
Ferguson has a long history of disciplinary charges with the FA.
In March 2011 Ferguson was hit with a five-match touchline ban and a £30,000 fine for questioning the integrity of referee Martin Atkinson after a match at Chelsea.
He did escape punishment earlier this season, however, despite confronting Mike Dean, assistant referee Jake Collin and fourth official Neil Swarbrick during the Red Devils' game against Newcastle.