Forest believe they were denied two or three clear penalties in the contest and accused the referees of bias.
A league statement on Monday read: "The Premier League was extremely disappointed to read the comments made by Nottingham Forest on social media yesterday, following its match against Everton.
"We note The FA has confirmed it will be investigating the club's statement.
It is never appropriate to improperly question the integrity of match officials, and the nature of these comments means the Premier League will also be investigating the matter in relation to the League's Rules."
Forest's initial tweet stated: "Three extremely poor decisions - three penalties not given - which we simply cannot accept. We warned the PGMOL that the VAR is a Luton fan before the game but they didn't change him. Our patience has been tested multiple times. NFFC will now consider its options."
The Reds then followed that up on Monday with this statement: "Nottingham Forest has today submitted a formal request to the PGMOL to release into the public domain the audio recordings between officials during yesterday's match against Everton at Goodison Park," a further statement read.
"The club has requested this be shared for three key match incidents - Ashley Young's challenge on Giovanni Reyna (24th minute), Ashley Young's handball (44th minute) and Ashley Young's challenge on Callum Hudson-Odoi (56th minute). We firmly advocate for the broader football community and supporters to have access to the audio and transcript for full transparency, ensuring the integrity of our sport is upheld."