The Everton Shareholders Association (EFCSA) have released a petition from the fans.
The club's supporters are furious with how owner Farhad Moshiri has been running the club.
Everton are once again locked in a relegation battle, having narrowly avoided the drop last season.
This week, supporter campaign NSNOW posted a statement on social media which read: "Like all fans, we are appalled by the Everton board's recent statements & actions, including the use of unattributed leaks to the media, that have resulted in considerable damage to the good name of the club & especially the fans. For the board there is no recovery from this.
"Farhad Moshiri has to accelerate plans to replace the board with competent individuals. He must start this process immediately. As well as media activity this week, we will announce details of our plans for a further peaceful sit in after the Arsenal game #ALLTOGETHERNOW".
The EFCSA account then posted a petition link, along with the statement:
"In our opinion our club continues to underachieve both on and off the pitch and the relationship the Owner and the Board have with the wider fan base is at an all time low.
"It is with great regret that the Association is asking fellow Shareholders and the wider fan base to sign our online petition calling for a vote of no confidence in the Board of Directors."
