Offensive chants could be heard coming from sections of the home support during Sunday's Premier League meeting at Kenilworth Road.
The FA have now strongly condemned the behaviour and pledged to continue to address the issue of tragedy chanting in English football, while reports suggest that Liverpool have also written to Luton seeking answers over how they plan to act.
The FA said in a statement on Monday: "We are aware of tragedy chanting during the fixture between Luton Town and Liverpool.
"We are seeking observations from Luton Town and further detail from the Police.
"We strongly condemn chanting of this nature and will continue to work closely with our stakeholders across the game, including the clubs, leagues, fan groups, and the relevant authorities to proactively address the issue."
Liverpool manager Jurgen Klopp said he had not heard the chants himself after the 1-1 draw at Luton that saw Luis Diaz's late stoppage-time goal cancel out an 80th-minute opener by Tahith Chong, but added: "Shame on everyone who said it."
Luton counterpart Rob Edwards said after the game: "I'll find out what was said. I haven’t heard a thing so I can’t comment.
"I don’t want to say the wrong thing right now. I don’t condone, we don’t condone that."
In a lengthy statement then released on Monday night, Luton said they were "saddened" by the reports of inappropriate chanting and "extremely disappointed that a small number of supporters soured the occasion with chants that may be interpreted as being in relation to tragedies that have affected Liverpool FC in the past".
The Hatters added that they "condemn any kind of chanting that knowingly seeks to divide" and that their "safety and security team launched an internal investigation at the earliest opportunity".
Luton also said they "wholeheartedly apologise" to anyone offended by the chanting and warned that those responsible could face stadium bans and potential criminal prosecutions.
In full, Luton's statement read: "Luton Town Football Club is saddened by reports of inappropriate chanting towards Liverpool supporters during yesterday’s Premier League fixture at Kenilworth Road.
"The atmosphere inside the stadium was electric for the most part of a pulsating game, in which Rob Edwards’ side came close to pulling off one of the results of the season.
"While we do not wish to dampen the atmosphere at our home ground in any way, we are extremely disappointed that a small number of supporters soured the occasion with chants that may be interpreted as being in relation to tragedies that have affected Liverpool FC in the past.
"The Club condemns any kind of chanting that knowingly seeks to divide, and our safety and security team launched an internal investigation at the earliest opportunity.
"What has quickly become evident is that a number of people may have taken part without knowledge that the words used were in relation to the Hillsborough and Heysel tragedies, and we see the route to persuading supporters not to repeat these chants in future is through communication and education.
"On this basis we are reviewing CCTV and media footage from the match, and will speak to witnesses to identify individuals who may have taken part. Any perpetrators could face stadium bans and potential criminal prosecutions.
"If anyone has information and video evidence that will support the investigation, please email firstname.lastname@example.org with the subject ‘Inappropriate Chanting’.
"The eyes of the world are upon us in the Premier League, which we are all learning quickly to adapt to, and we have to remind our supporters that you are all ambassadors of the Club and it is your responsibility to behave according to the rules of the ground. This includes understanding the songs that cannot be sung in line with the Love Football, Protect The Game campaign.
"On behalf of all at Luton Town, we would like to wholeheartedly apologise to anyone offended by the chants heard during yesterday’s match, and will continue to work with supporter groups to educate fans on chants that are classed as tragedy abuse by football authorities, the police and CPS."
The Premier League also condemned the chants, releasing their own statement which read: "The Premier League condemns the tragedy-related chanting heard at yesterday's match between Luton Town and Liverpool."We continue to treat this as an unacceptable issue and are committed to addressing it as a priority. Those found guilty of tragedy-related abuse face an automatic club ban and will be referred to the police."