The Sun says the ugly confrontation happened in training for tomorrow's match against Aston Villa.
Bullard - just back in action after 16 months out with a knee injury - caught Baird, 25, with a tackle.
Northern Ireland international Baird started arguing with his team-mate and then lashed out as Bullard, 29, laughed off the incident.
Furious manager Roy Hodgson, locked in a desperate battle for Premiership survival, separated the players.
Then he lectured the team about the need to pull together and said he would not accept such behaviour.
Baird now faces a club punishment as Hodgson is a strict disciplinarian.
The incident failed to dampen the squad's sense of humour as they joked that Bullard's badly-swollen nose made him look like the Elephant Man.