Everton have criticised the "preposterous arrogance" of the clubs involved in the European Super League, while Fifa boss Gianni Infantino "strongly disapproves" of the plans.
Arsenal, Chelsea, Liverpool, Manchester City, Manchester United and Tottenham are among 12 clubs who have agreed to join the proposed league.
"The backlash is understandable and deserved," an Everton statement read.
"We urge them all to consider what they wish their legacy to be."
The proposed tournament would see teams play one another in midweek games in an attempt to have more matches between the big-name clubs.
However, it has been heavily criticised by fans, pundits, football's governing bodies and members of the UK government.
"There is a lot to throw away for maybe a short-term financial gain for some," said Infantino, the president of world football's governing body.
"If some elect to go their own way, they must live with the consequences of their choice. They are responsible for their choice completely. This means you are either in or you are out. You cannot be half in and half out."ESL created to save football - PerezSuper League Q&A - what happens next?
In a lengthy statement, the Everton board said the six clubs were "tarnishing the reputation of our league and the game", accused them of "disrespect" towards fellow Premier League clubs and said they were "taking for granted and even betraying the majority of football supporters across our country and beyond".
The statement said: "This preposterous arrogance is not wanted anywhere in football outside of the clubs that have drafted this plan.
"On behalf of everyone associated with Everton, we respectfully ask that the proposals are immediately withdrawn and that the private meetings and subversive practises that have brought our beautiful game to possibly its lowest ever position in terms of trust end now."
The 14 Premier League clubs not signed up to the ESL will meet on Tuesday to assess the proposals and consider a response.Alan Shearer says the proposed European Super League is "plain and simply wrong"
Real Madrid president Florentino Perez said that the new league was needed to "save football" and help recoup losses caused by the coronavirus pandemic.
However, Everton have accused the six Premier League clubs of acting "entirely in their own interests".
"At this time of national and international crisis - and a defining period for our game - clubs should be working together collaboratively with the ideals of our game and its supporters uppermost," the statement continued.
"Instead, these clubs have been secretly conspiring to break away from a football pyramid that has served them so well.
"The self-proclaimed Super Six appear intent on disenfranchising supporters across the game - including their own - by putting the very structure that underpins the game we love under threat."Players not to blame for Super League - Liverpool boss Jurgen Klopp
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