Midfielder Pogba was the subject of discriminatory posts online after failing to convert a penalty in the 1-1 draw at Molineux on Monday.
It is the third high-profile instance in less than seven days of a footballer in England being targeted with racial abuse on social media.
Chelsea last week condemned "abhorrent posts" aimed at Tammy Abraham after the forward had a spot-kick saved in the UEFA Super Cup loss to Liverpool.
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 https://t.co/bzow073aTw— Harry Maguire (@HarryMaguire93) August 20, 2019
Reading striker Yakou Meite shared a screenshot of racially offensive messages he received in the wake of a late miss in Sunday's 3-0 victory over Cardiff City.
United pledged to take the "strongest course of action available" in a statement released to the media.
It read: "Everyone at Manchester United is disgusted by the racial abuse aimed at Paul Pogba last night and we utterly condemn it.
"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."
United's urging of social media platforms to stamp out racial abuse comes after Kick It Out lamented the "increasingly predictable" nature of the attacks.
"This is a call to action – we want to know what they are going to do to tackle this insidious problem," the anti-discrimination organisation said in a statement.