City midfielder Rodri escaped a handball call in the final minutes of the game at Goodison Park, with both the on-field referee and VAR opting against ruling in Everton's favour.
The Toffees were furious at the decision and have, according to The Athletic, demanded at the very least an apology for Frank Lampard and his team.
The report claims that Everton's chief executive officer, Denise Barrett-Baxendale, made the official complaint on behalf of the club to Premier League chief Richard Masters and chief football officer Tony Scholes earlier this morning.
The report adds that Barrett-Baxendale expressed 'grave concerns' over the standard of the refereeing in other Everton matches this season.
After the game, Lampard accused VAR official Kavanagh of 'incompetence at best'.
In an explosive outburst, he said: "We've lost a point by a professional who can't do his job right,' Lampard said. 'It's incompetence at best. At worst, who knows?
"That's Chris Kavanagh. It's completely on Chris. The referee [Paul Tierney] on the day knows it was a penalty. [But] he couldn't see it.
"The decision is incredible, incredible that loses us the opportunity to get what we deserved. It wouldn't have needed more than five seconds to know it was a penalty. He should have told the referee to give it or to look at it.
"They have not given that as a penalty when it strikes him on the arm, in an unnatural position. I have a three-year-old daughter at home who could tell you that was a penalty."
However, City boss Pep Guardiola believes the correct decision had been made. "It was offside," he said. "There was VAR and the action was offside. I didn't see the image."