The Football Association is understood to be against the Project Big Picture proposals, which would reshape professional football in England, while Premier League clubs are said to have serious concerns.
The proposals - drawn up by Liverpool's owners and backed by Manchester United - were revealed on Sunday and would change the voting structure of the Premier League as well as funding models for the English Football League and the FA.
The proposed changes would put the majority of power into the hands of the biggest clubs, ending the Premier League's current one-club, one-vote system.
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This power shift is understood to be opposed by the FA, which has the power to veto any fundamental changes to the Premier League thanks to the 'golden share' it was given when the league was created in 1992.
Despite the fact that the plans would see their power over the running of the Premier League increase, the rest of the big-six clubs - Chelsea, Manchester City, Arsenal and Tottenham - are also understood to have reservations.
The majority of the remaining 14 Premier League clubs have serious concerns over the proposals, with even West Ham - who would receive special status as one of the league's longest-serving teams - said not to be in favour.
Premier League clubs, who have already taken a financial hit during the coronavirus pandemic, are believed to be worried that the plans would negatively impact their accounts.
The proposal to reduce the top division from 20 to 18 clubs would have the effect of removing two home games from each clubs' calendar, as well as increasing the risk of relegation to the Championship.
Any changes to the structure of the Premier League require support from 14 of the 20 current clubs. As it stands, the proposals are unlikely to be backed by that number, meaning the FA will not be forced to decide whether to use its veto.
The Premier League shareholders are meeting this week - likely on Wednesday - while there is an FA council and board meeting on Thursday, with Project Big Picture likely to be high on the agenda at both gatherings.
'Like turkeys voting for Christmas'
Sky Sports News reporter, Kaveh Solhekol:
"My understanding of the situation is that the FA would not support these proposals in their current form. That is very important because the FA has a 'golden share' in the Premier League. Were these proposals to come up for a vote, the FA could veto them.
"What about support from Premier League clubs? My understanding of the situation is that Premier League clubs, by and large, are very lukewarm when it comes to their response to these proposals.
"You would expect the 'big six' to be in favour of them because it gives them so much power, but even amongst those clubs, there are significant reservations. I also don't think West Ham would back these proposals at all.
"My understanding of the situation is a lot of Premier League clubs think that voting for these proposals would be like a turkey voting for Christmas. Their finances would be hit. Their finances have already been hit by the pandemic.
"Under these proposals, the Premier League would be cut to 18 clubs, which would mean two fewer home games per season. Also, 25 per cent of the TV rights would be given to the EFL. Finally - and very importantly - because there would be only 18 clubs in the Premier League, there would be an increased threat of relegation.
"By and large, the majority of clubs in the Premier League are not in favour of these proposals."
Premier League: Proposals would damage English game
Following the release of Liverpool and Manchester United's Project Big Picture proposals, the Premier League issued a statement outlining its opposition.
It read: "English football is the world's most watched and has a vibrant, dynamic and competitive league structure that drives interest around the globe. To maintain this position, it is important that we all work together.
"Both the Premier League and the FA support a wide-ranging discussion on the future of the game, including its competition structures, calendar and overall financing, particularly in light of the effects of Covid-19.
"Football has many stakeholders, therefore this work should be carried out through the proper channels enabling all clubs and stakeholders the opportunity to contribute.
"In the Premier League's view, a number of the individual proposals in the plan published today could have a damaging impact on the whole game and we are disappointed to see that Rick Parry, chair of the EFL, has given his on-the-record support.
"The Premier League has been working in good faith with its clubs and the EFL to seek a resolution to the requirement for Covid-19 rescue funding. This work will continue."
A spokesperson for Boris Johnson made it clear on Monday that the Prime Minister also does not support the Project Big Picture plans.
The spokesperson said: "It is clear that this proposal does not command support throughout the Premier League, and it is exactly this type of backroom dealing that undermines trust in football governance."
'Plans extremely attractive to EFL'
While Project Big Picture has attracted plenty of criticism, EFL chairman Rick Parry has backed the proposals, saying they provide "long-term sustainability" for the clubs under his jurisdiction.
The plans would see EFL sides receive a £250m package, as well as 25 per cent of television deals negotiated by the Premier League.
Speaking on Sunday evening, Parry said: "This is a proposal to reset the long-term future of the English pyramid.
"This would produce long-term sustainability for all of our clubs. It would narrow the gap between the Championship and the Premier League. It would abolish parachute payments, which create a major imbalance within the Championship. Plus, there's a short-term package of immediate relief.
"It probably is the biggest idea since the formation of the Premier League. In terms of rebalancing the game, providing fairer shares for all, securing the financial future of the pyramid - for us, it's extremely attractive."
Parry is expected to meet with clubs from the Championship, League One and League Two on Tuesday to explain in detail how he believes Project Big Picture will affect them.
'Preston would welcome proposals'
Peter Ridsdale, the one-time Leeds and Cardiff chairman who is now acting as an advisor at Preston, can see why the proposals have been welcomed by some in the Football League - including Preston.
The prospect of 25 per cent of Premier League annual revenue being shared with Football League clubs annually, up from four per cent, represents a huge financial incentive.
However, he did question the merits of imposing a structure in the Premier League, concentrating power around the top six, when the teams that finish in those positions can and do change over time.
"If you take each sort of 10 year period you find, because of evolution and maybe different levels of investment, the clubs that are in the top six at a moment in time aren't the top six in 10 years' time or 20 years' time," Ridsdale said.
"So to try and put in a voting structure that almost guarantees that status, I would raise eyebrows were I to be in the Premier League - which of course I am not.
"From a Football League point of view, I can see why people would welcome it and we certainly would. It appears to me that there is potentially three times the amount of money on offer, once this is determined, on a seasonal basis."