Pogba was targeted after his second half penalty kick was saved by Wolves goalkeeper Rui Patricio during the Premier League game.
He is the third player, after Chelsea’s Tammy Abraham and Reading’s Yakou Meite, to be racially abused on social media in the past week after failing to score from the penalty spot.
In a statement, United said that those responsible for the abuse “do not represent the values of our great club.”
“Everyone at Manchester United is disgusted by the racial abuse aimed at Paul Pogba last night and we utterly condemn it,” the club said in a statement.
“The individuals who expressed these views do not represent the values of our great club and it is encouraging to see the vast majority of our fans condemn this on social media also.
“Manchester United has zero tolerance of any form of racism or discrimination and a long-standing commitment to campaigning against it through our #AllRedAllEqual initiative.
“We will work to identify the few involved in these incidents and take the strongest course of action available to us. We also encourage social media companies to take action in these cases.”
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Pogba’s United teammate Harry Maguire posted a tweet calling the abuse “disgusting.
“Disgusting. Social media need to do something about it… Every account that is opened should be verified by a passport/driving licence. Stop these pathetic trolls making numerous accounts to abuse people @Twitter @Instagram.”
United striker Marcus Rashford tweeted: “Enough now, this needs to stop @Twitter. Manchester United is a family. @paulpogba is a huge part of that family. You attack him you attack us all… @ManUtd.
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Kick it Out, a UK-based anti-racism charity, said the abuse was “unwarranted” and “vile.”
“The latest round of fixtures have seen unwarranted and vile racist abuse sent to players,” Kick it Out said in a statement.
“This time Reading’s Yakou Meite and Manchester United’s Paul Pogba. The number of posts such as these since the start of the season further highlights how discriminatory abuse online is out of control.
“Without immediate and the strongest possible action these cowardly acts will continue to grow.”
According to its records, Kick it Out says racist abuse increased by 43% last season, with 274 incidents compared to 192 the previous term.
In a statement sent to CNN, Twitter said it strongly condemned the “unacceptable behavior” of those responsible for the racist abuse and that the social media company had permanently suspended a series of accounts for violating its conduct policy.
Greater Manchester Police said it was liaising with United to provide support to the club and Pogba.