Pogba had a second-half penalty saved in the 1-1 draw with Wolves on Monday.
The 26-year-old is the third player in a week to be racially abused on social media following a penalty miss.
Another United player, Marcus Rashford, also expressed support for the France midfielder, saying “enough now”.
“Manchester United is a family. Paul Pogba is a huge part of that family. You attack him you attack us all,” England forward Rashford said on Twitter.
United said they are “working to identify” those responsible for the racist abuse directed at record-signing Pogba.
“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,” read a United statement.
“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.”
England centre-back Maguire also said he was “disgusted” by the abuse aimed at Pogba and suggested that social media users should have to verify their identity before opening an account.
In July, football’s anti-discrimination charity Kick It Out published a report that stated incidents of racist abuse increased by 43% last season, with 274 cases compared to 192 the previous season.
Following the abuse aimed at Pogba, the body said: “The number of posts such as these since the start of the season further highlights how discriminatory abuse online is out of control.”
It called the messages “unwarranted and vile racist abuse” and added “without immediate and the strongest possible action these cowardly acts will continue to grow”.
Kick It Out’s head of development Troy Townsend said those being “allowed to spout targeted racial hatred” were “festering in numbers due to lack of accountability and action”.
He added: “They are not all bots.”
Last week, Chelsea condemned “abhorrent posts” aimed at Tammy Abraham after his decisive penalty miss in the Super Cup against Liverpool.
Blues boss Frank Lampard said he was “disgusted” by the abuse aimed at the 21-year-old England forward, and like Maguire, called on social media companies to do more to stop players being targeted.
Then on Sunday, Reading striker Yakou Meite called out racist abuse sent to him on social media after he had a spot-kick saved.
Ivory Coast forward Meite, 23, came on as a substitute against Cardiff and missed a 91st-minute penalty.
Others fans quickly condemned the tweets containing racist abuse that were sent to Pogba – who won the penalty he then missed – shortly after full-time at Molineux on Monday.
Some tweets were later deleted while several accounts appeared to have been taken down.
Twitter’s terms and conditions say it “takes action against behaviour that targets individuals with hateful conduct”.