They are believed to have claimed that, having turned down vociferous Southampton appeals for a penalty, Clattenburg verbally abused midfielder Adam Lallana, and have requested that he not take charge of any of their fixtures until the matter is resolved.
The Professional Game Match Officials Limited (PGMOL), headed by former referee Mike Riley, is set to investigate the incident.
But Southampton assistant manager Jesus Perez says the club will not allow the controversy to have a negative impact.
Perez steered clear of the issue at a Friday press conference ahead of Saturday's FA Cup third-round tie at home to Burnley, although he said some news is expected in the coming days.
"I won't talk about this situation," said Perez. "I prefer to stay away from this and just focus on the game on Saturday.
"This type of situation doesn't help us in this moment. For this game, it's away, and we'll see in the next couple of days what's happening with this.
"I read it this morning. When you are in the games sometimes you can see or hear something, but history has led me to stay away from these things."
Saints focused on Burnley
And Perez insisted that the incident would not affect Southampton's preparations for the Burnley match.
"Our way always is, when the game is over, it's time to think about the next game," he said.
"So our full focus is to prepare well for the next game, then to ensure we're in best condition for the next match. That's our style since we came to England.
"It's not difficult for us, we focus and we put our players away from the problems and just think about the football."
Everton boss Robert Martinez says he is of the opinion that Clattenburg did a good job of refereeing the Goodison Park contest.
"That's the first time I've heard about that (Southampton's complaint)," said Martinez. "I'm quite surprised, I don't think that was the case.
"I thought Mark Clattenburg had a very good game in terms of controlling the players and showing a great manner."
Wenger fine with Clattenburg
Last season Clattenburg was accused by Chelsea of racially abusing midfielder John Mikel Obi during a fractious match against Manchester United at Stamford Bridge.
However, an FA inquiry ruled that there was no case to answer and Arsenal boss Arsene Wenger says he expects Clattenburg to be in charge for Saturday's FA Cup tie against Tottenham, despite this fresh controversy.
"I will be disappointed if Mark Clattenburg is not the referee tomorrow because, as long as you're not judged guilty, you're innocent," said Wenger
"It would be a very bad decision. Nobody has proven yet that he is guilty of what he is accused of.
"I personally have no problem with Mark Clattenburg refereeing the game tomorrow."