The FA is considering rule changes to prevent any future attempts to form a breakaway 'Super League'.
Fan protests were held after Arsenal, Chelsea, Liverpool, Manchester City, Manchester United and Tottenham agreed to join the breakaway league on Sunday night.
The backlash led to all six Premier League clubs withdrawing by Tuesday evening.
The other 14 Premier League clubs are divided on whether the 'Big Six' should face sanctions, Sky Sports News has been told.
One senior club official has said they want to pursue the possibility of punishments because they feel there is a clear breach of Premier League rules, and a precedent must be set to act as a deterrent to possible future breakaways.
Premier League Rule L9 says any member club needs prior written approval by the Board to enter a new competition, and this official says that rule has "patently been broken" by the rebels signing an agreement to join the proposed Super League.
An executive at another Premier League club has told Sky Sports News any sanction imposed on the 'Big Six', such as points deductions or fines would hurt the wrong people - the players, managers, and, crucially, the fans of those clubs.
"Our primary focus now is working to ensure that this can never happen again," said FA chief executive Mark Bullingham.
"We are exploring all options to prevent that, including legislation and changes to our rulebook, and nothing is off the table."
The UK Government has launched a fan-led review into models of ownership, governance, and the way the game is financed.
Tracey Crouch MP, who will chair the review, said: "Football means so much to so many people in this country and my review will be firmly focused on the fans.
"It will look closely at the issues of governance, ownership and finance and take the necessary steps to retain the game's integrity, competitiveness and, most importantly, the bond that clubs have with its supporters and the local community."
Manchester United boss Ole Gunnar Solskjaer: "I'm so happy that all the owners, all the clubs involved agreed that this was a mistake."
Manchester City boss Pep Guardiola: "Learn from that. You have to change something in the future. The Champions League was just for the champion that it was four [English] teams in the competition - maybe in the future it will change.
"They were thinking the best for the equality, and everyone had the opportunity to play the competitions that deserved to play. It's only after that, the people in charge, with wise and smart brains, to think what is the best for the future of football. Just be equality for everyone, when it's equality in the human rights and everything, society will be better."
Newcastle boss Steve Bruce: "I was like everybody else, and thankfully the 'Big Six' have taken a slapping, haven't they?
"We've got to guard against it because this thing must have been rumbling on for years, so we must guard against it so this threatened breakaway doesn't happen."
Leeds boss Marcelo Bielsa: "I am of the opinion that punishment is more for authority and shows that the quality or the efficiency of the rules was not up to standard."
Football's Civil War - Sky Sports News special
Join us at 5pm this evening for a two-and-half-hour Sky Sports News special show looking into how the breakaway Super League rocked football to its very core and asks what now for Britain's national game.
The show will include new interviews with Pep Guardiola, Gary Neville and Nemanja Matic and we will be joined by John Barnes, Alan Smith, Raphael Hongistein and others to cover one of the biggest stories in British Sport for decades.
Watch live on the Sky Sports News YouTube channel as well as Sky Sports News, Sky Sports Football and Sky Sports Premier League.