The pair - who are scheduled to meet at the O2 Arena on March 4 - had to be separated by security as they faced the media for the first time since their heavyweight showdown was announced last week.
After a press conference which saw both men verbally go back and forth for its entirety, tensions spilled over when they went face to face and Bellew feels that Haye's actions show that he is "very rattled" ahead of the highly-anticipated clash.
The 34-year-old told Sky Sports News: "We were talking, the security got a little bit too close and when David stepped up, he jammed his nose right up against my face, so I pushed him back to move him away from me and he punched me with a left hook.
"If that's the best he's got, he's in trouble on March 4 because I thought he hit harder than that, he definitely hits harder in sparring but is what it is, he's an idiot. I don't know why he's throwing a punch, I was literally trying to get him off me and get his face out of my face.
"He seems very rattled and annoyed, listen if he wants to take it to the streets than I am more than happy because there's only one winner on the streets with someone like me."
Bellew currently holds the WBC cruiserweight belt, while Haye is looking to secure himself a world title opportunity in 2017.