Allam has been looking into the possibility of re-branding Hull City as Hull Tigers, a move which has met with opposition from many of the club's supporters.
The idea took a further blow when the Football Association rejected the name change, with Allam now threatening to walk away from the club if he is not allowed to make his desired alterations.
Hull boss Steve Bruce takes his side down to face West Ham in the Barclays Premier League on Wednesday evening having done well in the club's top-flight return, currently sitting 12th in the table and with an FA Cup semi-final meeting with Sheffield United on the horizon.
Bruce's close friend Allardyce labelled the 'Tigers' issue a "distraction" from Hull's achievements.
And Allardyce is in no doubt that his owners would not pitch such a project to supporters.
"Owners are entitled to do what they want in circumstances regarding internal affairs with a club, whether they should choose a new manager or a different group of staff or whatever players they want.
"But I think when it comes to messing with the history and tradition of a football club they have to be very mindful of what it means to people.
"It means so much to people's lives to have that history, tradition, the name and the colours and what they stand for. It shouldn't be messed with and should be left well alone."
Sullivan and Gold are due to move West Ham out of Upton Park and into the Olympic Stadium in 2016, in a significant upheaval given it means leaving what has been the club's home for over 100 years.
However that is a switch which has the overwhelming backing of West Ham's fans, with 85 per cent of those polled in May 2013 giving it their support.
West Ham face Hull on the back of three consecutive league defeats, with Allardyce bemoaning refereeing decisions in each game.
The Sun newspaper spoke to former professional official Mark Halsey who highlighted that Andre Marriner, who mistakenly sent off Arsenal's Kieran Gibbs after Alex Oxlade-Chamberlain had committed deliberate handball in the 6-0 loss at Chelsea, may have been tired following a Europa League commitment on Thursday evening.
Mason had also been on European duty before West Ham's match against the Red Devils, and Allardyce revealed he has been against referees not having a long enough break between matches for some time.
"I broached this subject many years ago when I was at Bolton," he said.
"We had an official for a game at 12:30pm on a Saturday after he had been in Europe and I knew he was in Europe because I had watched the European game and watched him referee it.
"We came back and found we had that referee and I said that wasn't right all of those years ago, but obviously it hasn't changed the opinion of the Premier League or the PGMOL [Professional Game Match Officials Limited] but it should do.
"Lee Mason was in Europe as well, he lives in Bolton. So he has gone out, got back to Bolton, then had to travel down to us to referee a game on Saturday.
"It is not a criticism of the referee, it is a criticism of the system for me and I think it is a very, very good criticism.
"Tottenham's worst results have come from the game after playing in Europe and that is being able to change the team when you know you can't play the same team.
"Referees can only be fatigued before they start that game when they have done not just done a journey but done another game before it, it is plain and simple.
"The problem we have is that we should have enough referees so that doesn't matter and I don't think we have enough referees to deal with the demand, sadly."
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