Liverpool say racist abuse on social media "cannot be allowed to continue" after Trent Alexander-Arnold, Naby Keita and Sadio Mane were targeted.
The Premier League club said they will work with relevant authorities to try to identify and prosecute offenders.
Liverpool, however, added it "will not be enough until the strongest possible preventative measures are taken".
In a statement, the club condemned the abuse as "utterly unacceptable".
"Once again we are sadly discussing abhorrent racial abuse the morning after a football game," they said after Tuesday night's Champions League quarter-final first leg defeat by Real Madrid.