Most fans feel the addition of Premier League academy sides to the EFL Trophy "undermines the integrity of the competition", according to the Football Supporters' Federation.
Sixteen category one academy sides will play League One and Two clubs in a bigger 64-team competition in 2016-17.
The change to the former Johnstone's Paint Trophy is a one-season trial.
"The reaction from the overwhelming majority of fans to the news has been incredibly negative," said a statement.
"Fans have told us that it undermines the integrity of the competition, and feel that 'their' competition, and the chance of a potential day out at Wembley (for the final), is being taken away from them by Premier League clubs."
To accommodate for the additional teams, the current knock-out format will be replaced by 16 regional groups of four for the opening round.
The proposal to change the competition, which is yet to gain a sponsor for next season, was approved at the English Football League's annual general meeting in Portugal last week.
It is understood that Premier League clubs are offering an extra £1m in prize money.
Those in favour
Since the announcement by the EFL, clubs have been issuing statements to clarify their position to supporters.
Walsall, whose only ever appearance at Wembley came in the 2015 final, said they voted in favour as "the competition's appeal is declining significantly in terms of interest from football fans, clubs and broadcast partners".
The Saddlers added that they did not see the change as a "long-term solution" but it was the only option put forward to rejuvenate the competition.
Meanwhile, Oxford United chairman Darryl Eales was against the proposal, but voted in favour "to support the League's appeal to explore ways to secure the cup's future".
Some fans have been against the changes over fears Premier League 'B teams' could be added to the League, especially given the proposal to expand the league structure from 92 teams to 100, with an extra division.
However, Exeter chairman Julian Tagg, another who voted for the changes to the EFL Trophy, said: "This is not a precursor to B teams entering the league structure, which Exeter City would be against should it ever be proposed."
Among those to speak out against the changes have been League Two side Luton Town, who voted against the decision "but a democratic process was taken so (they) will now be supportive of the new competition".
Hatters chief executive Gary Sweet added: "Unfortunately, the old JPT was in a state of flux so some change was indeed necessary to avoid losing the competition altogether."
AFC Wimbledon, promoted to League One via the play-offs last term, "strongly opposed" the proposals at the meeting in Portugal.
Accrington Stanley said they were "vociferously against the proposals".
"When you break it down there's no financial gain for a team like Accrington Stanley," said managing director David Burgess. "The financial rewards only really kick in when you get to the quarter-final, semi final and final stages.
"We don't feel there would be a benefit even to getting one of the bigger clubs like Manchester United, Manchester City, Liverpool or whoever to the Wham Stadium.
"Accrington played Manchester United here in the Lancashire Senior Cup here a couple of years ago and there were a few hundred people here. There's no real enthusiasm for under-21s football or reserve-team football or whatever you want to call it."
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