The Blues said people making “unacceptable” chants were either prevented from entering Stamford Bridge or ejected during the game.
“This behaviour will not be tolerated,” said Chelsea on Tuesday.
“Those supporters will be banned from attending matches at Stamford Bridge in future.”
Chelsea added that they have “passed on the details of those supporters” to Manchester United so they can “take appropriate action”.
“Chelsea is a diverse and inclusive club and will always take firm action against all forms of discriminatory behaviour,” they added.
United won the match 2-0 through goals from Anthony Martial and Harry Maguire.