The Spurs midfielder allegedly made a threatening gesture toward the ex-Arsenal man.
Off-the-field incidents marred an otherwise excellent match as the Blues came from two goals down to deny Spurs a victory which would have kept them in the Premier League title race, with Leicester City crowned champions as a result of their draw.
Straight after the match, a melee ensued at the entrance to the tunnel where players and staff members for both sides were involved, resulting in 69-year-old Chelsea boss Guus Hiddink being accidentally pushed to the ground.
Ex-Arsenal man Fabregas is alleged to have been flicking Spurs' players in the groin during the mess, and Dier, according to The Mail, then made the threatening gesture to the 29-year-old Spaniard.
The former Arsenal midfielder is also accused of shouting profanities in Spanish at Tottenham's coaching staff, as well as winger Erik Lamela in an ugly end to a wonderfully exciting match in West London.
On the pitch, the title is gone for Spurs, but one more victory would effectively guarantee a higher league finish than Arsenal for the first time in 21 years.