The Egyptian-born businessman took over at the KC Stadium outfit in 2010, but has angered fans with his plans to rename them Hull Tigers.
A section of supporters, going by the name City Till We Die, put up a banner during last week's defeat to Crystal Palace which read "We are Hull City".
They also chanted 'City till we die' throughout, and Allam has said those fans can "die as soon as they want", while warning them that he will not waste time in putting the club up for sale if they continue.
"I don't mind 'City till we die'", he told the Independent on Sunday "They can die as soon as they want, as long as they leave the club for the majority who just want to watch good football.
"How can they call themselves fans, these hooligans, this militant minority, when they disturb and distract the players while taking away the rights of others to watch the football, and of companies who have paid good money for advertising?
"If they want to express their feelings they are free to do so, either outside the stadium or pay to take space. Seriously, they are welcome to talk to the stadium management about buying a space for a permanent banner, 10 times as big if they want. I am a supporter of democracy. I would have no issue with that.
"I'm a simple man. Do they want me to stay? If it's 'No thank you', fine, in 24 hours the club is for sale, I do not put in one more pound and hopefully things happen quickly."
Hull host Liverpool in the Premier League on Sunday.