Liverpool say racist abuse on social media "cannot be allowed to continue" after Trent Alexander-Arnold and Naby Keita were targeted.
Both had racist emojis sent to them on Instagram in the lead up to Tuesday's Champions League tie with Real Madrid.
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.
"As a club, we will offer our players any and all support that they may require.
"The current situation cannot be allowed to continue and it is incumbent on all of us to ensure it does not."
Former Arsenal and France striker Thierry Henry last week removed himself from all social media because of racism and bullying across platforms.
In an interview with BBC's Newsnight programme, the 43-year-old said "enough was enough" and that he had to make a stand against racism on social media.
On Tuesday, Brentford striker Ivan Toney also received racist abuse on Instagram.
The Championship club called the abuse "spineless".
The UK government has previously threatened social media companies with "large fines" which could amount to "billions of pounds" if they fail to tackle abuse on their platforms.
Facebook, which owns Instagram, said in February that tougher measures would be taken to tackle the issue.