The British rivals will go head to head in London on March 4.
The British rivals, who will go head to head in London on March 4, had to be separated and marshalled into different rooms having spent the previous 20 minutes trying to shout over each other.
Former world champion Haye, 36, who cut an angry and aggressive figure throughout, also insulted promoter Eddie Hearn and Bellew's trainer David Coldwell.
He and Bellew, the WBC cruiserweight champion who is stepping up to heavyweight for the first time, exchanged barbs before Bellew shoved Haye and they nearly came to blows.
Bellew had accused Haye of struggling financially and spoke of the difficulties involved in negotiating with him.
"You could be fighting for the heavyweight championship of the world, but you're fighting me," said the 34-year-old Bellew.
"And you know why? For the money. You're skint."
Haye fired back: "Let's see who's round your hospital bed on March 4. You're putting this guy in such a dangerous situation.
"I end the fight when I want. It's completely up to me, you've got no say."
Haye has previously been involved in violence at a press conference when he brawled with Dereck Chisora in 2012.