Manchester City star Sterling was near the touchline and heading for the corner flag as he was abused by a fan, who appeared to shout ‘f**king black c**t, f**k off’.
Another shouts ‘f**k you, you c**t’.
The disguting video was posted on Twitter by John Sproull.
BT Sport pundit Jake Humphrey responded to Times journalist Matthew Syed who also tweeted about the incident.
He said: “Hi @matthewsyed. Just been speaking to the production team.
“The footage is all over social media & we’ll make sure the relevant people are sent the footage directly from us.
“If indeed it is proved to be racist abuse then I hope it’s a lifetime ban. No place for it in this world.”
Syed, who also posted a clip of the video, originally tweeted: “This guy, far right, blue jacket, glasses, screaming what looks like vile racial abuse. Kick him out @ChelseaFC @premierleague @kickitout. Does anybody know him? Not just awful for Sterling but other fans, not least the black guy behind him.”
It is another bleak reminder of the racist issues in football after a Tottenham supporter threw a banana skin at Arsenal striker Pierre Emetic Aubameyang last week.
That fan was given a lifetime ban by Tottenham and it remains to be seen if this Chelsea fan will be identified and hit with a similar punishment.
Chelsea beat Manchester City at Stamford Bridge, allowing Liverpool to maintain top spot after their win over Bournemouth earlier in the day.
It was City’s first loss in the league this season after 16 games.
Goals from N’Golo Kante and David Luiz secured the three points for Chelsea that sees them go back up to third in the table.
After the game goalscorer Luiz said: “It was a great game. I think we are playing against the best team in Europe at the moment. They have great players and are playing very well.
“We had to suffer but then we took our chance to score. It went well, our plan. We have had some bad results but tried to improve.
“All the team did great. We keep fighting but keep our feet on the ground.”
And Kante said: “They are a very good team. We suffered together then afterwards we didn’t give up and got the two goals. We are happy with the win.
“The most important thing is to give my best on the pitch. I am happy to be on the scoresheet.”