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Hayman (39), who played 45 tests between 2001 and 2007, was involved in a bust-up with unnamed players at the start of the week and has been relieved of his duties as forwards coach, the club said in a statement.
He was not involved with any training sessions in the build-up to Pau's 40-23 victory over Bordeaux-Begles - the team's first league win since October.
Hayman has been in charge of the Top 14 side's forwards since 2016 after winning the French title and three European Cups as a player with Toulon.
Pau president Bernard Pontneau confirmed former Brive coach Nicolas Godignon will take over from Hayman but insisted the former front-rower will remain involved with the club.
"Carl is moving on to a personal project which we will support. He will stay with the club until the end of June. Nicolas Godignon will join us next week," Pontneau told French broadcaster Canal+.
Pontneau didn't give details about the altercation but said the decision to sack Hayman "touches me personally".
"I have a lot of affection for him, for the person that he is. It'd a feeling shared by the entire club," Pontneau said.
Fellow New Zealanders Simon Mannix, Conrad Smith and Paul Tito are also part of the coaching set-up at Pau, while the club has seven New Zealand players on their books - including World Cup winner Colin Slade.
Hayman made his test debut in 2001 but only established himself as the first-choice tighthead under Graham Henry in 2004.
He was only 27 when he signed with Newcastle for a then-record $1 million annual salary with Newcastle. He signed with Toulon three years later.
Last month, Pontneau slammed New Zealand Rugby after All Black flanker Ardie Savea backed out of a deal to join the club, with the 25-year-old Hurricanes flanker deciding to remain in New Zealand at the last minute.