They sit ninth in the Top 14 standings and are slipping further away from their goal of qualifying for the Heineken Cup.
After a meeting from the board of directors on Monday and a vote amongst the players yesterday, club president Thierry Pérez has decided to relieve Nourault and Hyardet of their duties.
The duo will have to undergo interviews if they wish to remain at the club and will no longer be involved in first-team activities if they are kept on.
Perez and Nourault have a close relationship, with the latter having been in charge at Montpelllier for a decade and helping to make Montpellier a Top 14 force after previously being a Pro D2 outfit.
However, the president admitted that results and the field had to be put before friendship and that action was therefore needed sooner rather than later.
"A new page has turned for the club," said Pérez.
"Friendship is one thing, but reality and the club's ambitions are others."
"It's above all a question of atmosphere. The players no longer have confidence in them."
No appointment of a head coach will be made this season, with team manager Denis Navizet selecting the side along with Pascal Mancuso.
Training will be taken by Jean-Philippe Lacoste (attack), Warren Britz (defence) and Didier Bès (scrums, forwards).