Bellamy marched towards the man, who staged a solo pitch invasion, and shoved him in the face before the fan was led away by match stewards at the Manchester derby game.
The man ran from the Manchester United home section as Manchester City took the kick-off following Michael Owen's late goal in United's 4-3 victory.
Neither the fan nor the Wales international have made a complaint to police about the clash.
A Greater Manchester Police spokesman said: "At the end of the derby between Manchester United and Manchester City at Old Trafford, a member of the public ran on to the pitch, and was detained by police and security staff.
"During this time, there was an altercation between the man and a player from Manchester City FC.
"No formal complaint has been made by either party but police are continuing to make inquiries."