A breakaway league was suggested "as a threat" by the organisers of Project Big Picture, says Football Association chairman Greg Clarke.
Project Big Picture wants to reduce the Premier League from 20 to 18 clubs and scrap the Carabao Cup.
The EFL would also get 25% of all future TV deals, plus £250m.
Fan groups of the ‘big six’ teams - Arsenal, Chelsea, Liverpool, Manchester City, Manchester United and Tottenham - have criticised the plans.
Some EFL clubs said support for the plans was "almost unanimous" among their member clubs - although that was later disputed.
FA chairman Clarke said he was involved in initial discussions with the backers of Project Big Picture but then withdrew.
“Change must benefit clubs, fans and players; not just selective balance sheets,” said Clarke.
“With the knowledge of senior board members and our CEO, I participated in the early stages of discussions.
“However, when the principal aim of these discussions became the concentration of power and wealth in the hands of a few clubs with a breakaway league mooted as a threat, I discontinued my involvement and counselled a more consensus-based approach involving all Premier League clubs.
“Our game needs to continually seek to improve, but benefits need to be shared.”
The proposals, led by Liverpool and Manchester United, became public earlier this week and have split opinion.
EFL chairman Rick Parry said it was “in the best interests” of football in this country and praised both clubs for coming up with the plan.
"This is two of our great clubs showing leadership and exercising responsibility," said Parry. "The message from Liverpool and Manchester United is that they do genuinely care about the pyramid."
However, the Premier League thought the plan "could have a damaging impact on the whole game" and the Department for Culture, Media and Sport said it was "surprised and disappointed" by "backroom deals being cooked up".
A West Ham insider told BBC Sport they were "very much against" it and a spokesman for Prime Minister Boris Johnson said the plans would "undermine the trust in football's governance”.
Clarke said discussions should continue, adding: “We, the FA board and council, have to ensure any changes would be to the long-term benefit of the whole of football.
“We have substantial controls to help ensure that the best interests of the game are served by any new proposals.
“In these difficult times unity, transparency and common purpose must override the interests of the few.
“Let’s continue to work together to determine what is best for English football, with full dialogue between all key stakeholders.”
The 'Project Big Picture' proposals
- The Premier League cut from 20 to 18 clubs, with the Championship, League One and League Two each retaining 24 teams.
- The bottom two teams in the Premier League relegated automatically with the 16th-placed team joining the Championship play-offs.
- The League Cup and Community Shield abolished.
- Parachute payments scrapped.
- A £250m rescue fund made immediately available to the EFL and 25% of all future TV deals.
- £100m paid to the FA to make up for lost revenue.
- Nine clubs given 'special voting rights' on certain issues, based on their extended runs in the Premier League.
‘We do not support the proposals’
On Tuesday, a joint statement from supporters' groups of the ‘big six’ teams said: “The fans we represent are fortunate to support clubs that regularly secure the largest financial revenues in the Premier League. But all of us understand that football doesn’t work in isolation - it’s a family.
“It requires a fair share of resources to ensure that the Premier League is competitive to watch and that the lower leagues flourish as part of our national game.
“While the six clubs we support are widely reported to be the instigators of Project Big Picture, it is important we state very clearly we do not support the proposals in their current form.
“We are totally opposed to concentrating power in the hands of six billionaire owners and departing from the one club, one vote and collective ethos of the Premier League. This part of the proposal must be dropped immediately if other elements are to be given serious consideration.
“We welcome the government reiterating its plan for a supporter-led review into football governance. This needs to happen as a matter of urgency and supporter groups must be consulted."
'Almost unanimous support in EFL'
Leading figures from EFL clubs in the Championship, League One and League Two said there was almost unanimous support for the proposals during individual league meetings on Tuesday.
Jez Moxey, chief executive of League One side Burton Albion and a member of the EFL board, said: "I'd say in League One, [support is] unanimous."
Paul Scally, chairman of League One's Gillingham, said: "I don't think anyone spoke badly about it. There were concerns over certain aspects of it but the principle of it, as I've believed all along, has been very sound. I think it's welcomed.
"There is obviously more discussion to be had, but clubs aren't looking at selling their souls for £250m, this is not about taking the money, whatever the cost.
"This is about the future sustainability of the English Football League and the pyramid of football, of which the Premier League play such a big part. We would be very much hopeful that this can come to a positive conclusion."
Nigel Travis, chairman of League Two's Leyton Orient, said: "I'd say in League Two it was probably 23 out of 24 clubs [in favour]. One club raised a number of interesting issues."
He added: "Apart from a couple of clubs throughout the league we are totally behind it. It is to make football better and sustainable for all of us. Think about the income it brings into government, think about what it does for our communities."
Peter Ridsdale, the owner's representative at Championship side Preston North End, said: "At the Championship meeting today, whilst there wasn't a vote, there were no dissenting voices on the conference call.
"There were some questions raised, and I think fairly, because it was leaked, and because we've only seen certain elements of the discussions there will be points that individual clubs would want clarification on or the ability to discuss, but in terms of the broad principles of it there were no dissenting voices at the Championship meeting today."
However, the claims of almost unanimous support for the proposals were later disputed by other clubs.
Representatives of Accrington Stanley, Lincoln City and AFC Wimbledon said the comments were not an accurate reflection of the level of support for Project Big Picture.