Monkey emojis were posted in the comments underneath photos of both players in the aftermath of the game.
Facebook, who owns Instagram, has told Sky Sports they are investigating the incident.
"Once again we are sadly discussing abhorrent racial abuse the morning after a football game. It is utterly unacceptable and it has to stop," a Liverpool spokesperson said.
"LFC condemns all forms of discrimination and we continue to work with our inclusion partners through our Red Together initiative to campaign against it.
"As a club, we will offer our players any and all support that they may require. We will also work with the relevant authorities to identify and, if possible, prosecute those responsible. "We know that this will not be enough until the strongest possible preventative measures are taken by social media platforms and the regulatory bodies which govern them.
"The current situation cannot be allowed to continue and it is incumbent on all of us to ensure that it does not."
Facebook said in February it was "horrified" by the abuse footballers received on social media.
Manchester United quartet Marcus Rashford, Anthony Martial, Axel Tuanzebe (twice) and Lauren James as well as West Bromwich Albion's Romaine Sawyers, Chelsea's Reece James and Southampton's Alex Jankewitz have been victims of racist abuse online in recent weeks.
The Premier League has put a plan in place to eradicate racial prejudice, with one area of focus making it easier to report online abuse.