The 33-year-old played the full 90 minutes as the Red Devils were beaten 3-0 by Barcelona in the UEFA Champions League quarter-finals on Tuesday, meaning they were eliminated 4-0 on aggregate.
Per BBC Sport, United and said they were "working to identify individuals involved" in order to take the "strongest possible action."
On the pitch, it was a challenging night for Young and his team-mates in the United defence as they were pulled apart by the brilliant Lionel Messi.
Messi opened the scoring with a fantastic early goal before doubling his team's advantage when his tame shot somehow slipped under David de Gea in the United goal. Philippe Coutinho scored from distance in the second period to seal the result.
The abuse suffered by Young comes at a time where racism in football has been thrust into the limelight.
Tottenham Hotspur full-back Danny Rose told reporters in early April that he "can't wait to see the back of football" over frustrations with governing bodies' inability to tackle racist abuse.
On the weekend following the publication of Rose's quotes, Championship clubs Wigan Athletic and Brentford received allegations of racist abuse, with both cases referred to police. Wigan full-back Nathan Byrne was sent racist abuse online, while a Brentford season ticket holder was arrested after allegedly racially abusing Derby County's Duane Holmes.
Rose was one of a number of England players subjected to racist abuse in the win over Montenegro in late March, while other recent incidents have seen Moise Kean of Juventus and Chuks Aneke of MK Dons racially abused during games or on social media.
Young has been operating as a full-back in recent years, having initially arrived at United from Aston Villa as a winger. Mark Ogden of ESPN FC thinks his performances are suffering at the moment as a result of poor recruitment:
Kick It Out @kickitout
Yet another black player, this time Ashley Young, targeted with racist abuse on social media after tonight’s Champions League game. Again, we’re left asking @TwitterUK the same question - when will you take serious action to tackle the rampant discrimination on your platform? https://t.co/WMsO5GJ3Ey
The England international has been a useful player for the Red Devils since joining in 2011, winning the Premier League title in 2012-13. In the years since, he's also won the UEFA Europa League, FA Cup and League Cup.
In February, it was confirmed the veteran has signed a new contract with United, keeping him at Old Trafford until the end of next season.