The Daily Mail says violence erupted about 10 minutes after the ground emptied as Patrice Evra, Paul Scholes, Ji-sung Park, John O'Shea, Gerard Pique, Danny Wellbeck and Gary Neville were put through some gentle training drills.
It is understood that United players took exception to being told they could not run on a muddy part of the pitch, even though Premier League rules say the playing surface must be made available to the opposition.
A heated exchange then took place, during which one United player allegedly abused Chelsea's head groundsman and a member of the Stamford Bridge groundstaff told O'Shea: "I'll stick my pitchfork up your a**e."
The FA and Premier League are beginning their enquiries into the scenes that saw defender Evra and United team-mates fighting with Chelsea groundsman Sam Bethell on the Stamford Bridge pitch as they warmed down following the London club's 2-1 win on Saturday.
United have asked for a copy of Chelsea's CCTV footage. Both clubs are likely to be fined if an FA probe proves staff or players were to blame. Bethell said: "It was handbags. I've not lost my job."