The Black Cats are working with relevant authorities and say they will "take the strongest possible action available to us".
In a strong statement, Sunderland describe the abuse aimed at Mundle as "abhorrent" and say it was carried out by "multiple individuals". Sunderland say the offenders "are not welcome on Wearside".
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Mundle came off the bench for the Black Cats in Sunday's defeat to Fulham. The winger missed a chance in the second half and Regis Le Bris also revealed after the game that the winger had picked up an injury. Mundle was pictured looking devastated with his head in his hands after the game.
Sunderland released a statement later on Sunday evening.
It read: "Sunderland AFC is appalled by the vile online racist abuse directed at Romaine Mundle following today’s Premier League fixture against Fulham.
"The abhorrent behaviour displayed by multiple individuals is unacceptable and will not be tolerated by the club under any circumstances.
"There is no place for racism in our society, and we stand with Romaine, who has our full support.
"The club is actively working with the relevant authorities and online platforms to identify those responsible, and we will take the strongest possible action available to us.
"These individuals do not represent Sunderland AFC, our values, or our community - and they are not welcome on Wearside."