Footage emerged this morning of Moyes, speaking in the aftermath of a draw with Burnley on March 18, telling BBC reporter Vicki Sparks to “be careful” after asking if the presence of Sunderland owner Ellis Short had put pressure on him.
Moyes said “no”, adding that Sparks “might get a slap even though you’re a woman”.
The FA will write to Moyes to ask for his observations of the incident.
The 53-year-old Scot has said he “deeply regrets” his comments.
Speaking in his press conference on Monday he told reporters: "I have said that I regret it. I have spoken to the girl, who I apologised to, and she accepted it."
A spokesperson for the BBC said: "Mr Moyes has apologised to our reporter and she has accepted his apology."
Shadow sports minister Dr Rosena Allin-Khan, the MP for Tooting, called on Moyes to resign for his “sexist threats”.
She wrote on Twitter: “This is disgraceful. David Moyes cannot get away with these sexist threats - the FA must take action immediately.”
Moyes insisted he had "never" considered resigning over his comments, adding that his actions were "out of character".