The club's owners levelled the accusation at the FA in a statement released on Tuesday, 24 hours after it was revealed the FA's membership committee had recommended rejection of an application to rename the club 'Hull Tigers'.
A decision will be made at a meeting of the FA's Full Council on April 9, but Allam has vowed to put the matter to season pass holders in the hope of backing his proposal with a resounding 'yes' vote.
However, the City Till We Die campaign group is convinced supporters, many of whom have reacted angrily to the move to alter the club's name after 110 years, will not give him what he wants.
Spokesperson Mark Gretton told Press Association Sport: "There have been so many polls and every one of them has voted against a name-change.
"Every reputable poll that has been done, whether by a fans' group, the Hull Daily Mail or the official supporters' club, they have all voted against a name-change.
"That said, asking the fans is a good idea. We suggested this back in November.
"We wrote to him (Allam) about it, but he ignored us and in the press, that was when he started with his 'no one tells me how to run my companies'.
"We assumed he'd gone quiet on this. That would have been a very sensible thing to do back then so whatever the result, the pass holders can have a say in this decision."
City's statement confirmed that pass holders will be balloted on the matter, which Allam believes is vital if he is to make the club more marketable, with details of the vote due to be released within 48 hours.
The statement said: "The club is disappointed with the timing of yesterday's announcement made by The FA.
"We feel we have been prejudiced right from the outset. The FA introduced a new policy to handle our application on the back of consultations with the City Till We Die group, and we therefore feel that our application was already pre-judged.
"This feeling has been reinforced with yesterday's announcement, as the club feels this will further prejudice the forthcoming ballot of season pass holders.
"However, this is now the time for the silent majority to come forward and support the club's aspirations. Details of the ballot will be released to season pass holders within the next 48 hours."
Gretton was puzzled by the reaction, but heartened by the club's apparent failure to make its case with the FA.
He said: "I honestly don't know why they would say that. It's a very odd thing to say because as I understand it, there's been no decision made yet.
"There's been a recommendation which has to be ratified by the FA, so it does seem tactically a very odd thing to do, to accuse the FA when they is still a decision to be made."
Allam has threatened to walk away from the KC Stadium if he does not get his way.
However, members of the fans' group are convinced there is a way forward.
Gretton said: "We have been broadly happy with what's happened under the Allams' ownership of the club and we would like this to be finished. That's why we welcome a season ticket holders' poll.
"If we can get a popular vote against the name change, we are not going to crow about it - we have no need to.
"Hopefully, he would just abandon the plan and we could say 'well done for being the bigger man'. We'd be delighted about that and we could start afresh."
Meanwhile, Tigers striker George Boyd has been charged with misconduct by the FA for allegedly spitting at Manchester City keeper Joe Hart.
Boyd has until 6pm on Wednesday to respond to the charge.