The United boss could be in hot water with the governing body after turning his ire on the officials following United's loss in the first leg of their semi-final against Gus Poyet's side at the Stadium of Light on Wednesday night.
Moyes was not impressed with referee Andre Marriner's decision to award a penalty, which was converted by Fabio Borini, for a challenge by Tom Cleverley on Adam Johnson.
When contacted by Perform on Wednesday, the FA confirmed they are looking into the former Everton manager's comments and they will also contact both clubs before deciding whether to act after flares were lit by United fans.
Moyes is under pressure in his first season at Old Trafford, with United seventh in the Premier League and out of the FA Cup following their defeat to Swansea City on Sunday.
But he claimed the Premier League champions have been on the wrong end of some poor decisions.
"I feel we are having to play the opposition and the officials," said Moyes after United's latest defeat.
"They gave a free-kick on the first goal that I couldn't believe. Maybe I have got to understand that’s what happens at Manchester United - they are certainly not giving us much at this moment in time.
"For the penalty, the linesman gives it, but he can't see through Patrice Evra. The referee is looking right at it and is in a very good position - not too close, not too far away, but he turns around. The linesman gives it.
"I just think it is incredible when you think we didn't get one the other week for the (Hugo Lloris) challenge against Tottenham. It’s hard to take, it really is."
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