The British boxers, who have had a long-running feud, had to be pulled apart by security while facing one another for pictures.
Verbal digs were thrown by each fighter during the press conference while they discussed their upcoming fight, which will be held at the O2 Arena in London on March 4.
Haye told Bellow: "Whatever I do now in my life is irrelevant because on fight night, you're going to get knocked out. I could go clubbing every night.
"From now until the fight - get smashed out of my mind every night, and I'll still knock you out. I will turn up, drunk out of my head, and you'll get knocked out."
In response, the Liverpudlian boxer, who is the reigning world cruiserweight champion, hit out at Haye's demands.
He said: "I don't fear him. He's got this thing that I'm scared of him. All those demands, all those stipulations you put in. 'I want the ring this size, I want the mat that size'... Listen mate, I'm not going to run from you. I know you believe I'm going to run.
"All these stipulations he put on, I'm surprised he didn't ask me to carry him to his seat in the press conference, to be honest. Do you want me to do your hair as well? The man is unbelievable. My demand was - let me pick my own gloves and get in the ring. He's unbelievable - the biggest diva in boxing I have ever seen."
Haye returned to the ring earlier this year after almost five years away from the sport.