Brewster, who joined United from Liverpool in October in a club-record deal, shared a screenshot on his own Instagram story of the message, which contained the N-word.
"Overrated n*****," it read. "You're s***, poo boy."
The account the message was sent from is in the name of 'James', who appears to be a Chelsea supporter with a profile picture of Mason Mount. United travel to Stamford Bridge this weekend in the quarter-finals of the FA Cup.
Brewster shared the message with the caption: "I bet you this kid is about 12".
And United today said in a statement: "As a club we applaud Rhian for bringing this disgusting message to light.
"We will now support him and work with him, the Premier League and all the relevant authorities to ensure the person behind this post is brought to justice."
A number of high-profile footballers have been targeted with racial abuse on social media recently, and a 22-year-old man appeared in a Scottish court on Wednesday charged with sending a racist message to Middlesbrough's Yannick Bolasie.
Instagram has recently said that it is committed to tackling online abuse, and will remove the accounts of people who send abusive messages. But the profile of 'James', at the time of writing, was still active.
David McGoldrick, Brewster's teammate at Bramall Lane, was also targeted with racist abuse on Instagram after scoring twice against Chelsea last season.