The Glasgow club said the employee, who is now dead, had worked at Partick for two years.
When the alleged abuse was raised in 1992, he was dismissed "with immediate effect", the club said.
Police Scotland and the Scottish Football Association have been informed.
In a statement, Partick Thistle said: "As far as Thistle's current management can ascertain, no other allegations were made to the club at that time with regard to him or any other employee.
"The club has contacted the SFA and Police Scotland to advise them of what they know to date and will fully comply with any investigation."
"In the meantime, Partick Thistle would urge anyone who may have been affected to contact Police Scotland, the SFA, the club or the NSPCC helpline (0800 0232642) set up to support and advise victims of abuse.
"In the last 24 hours, we have been approached by an anonymous individual asking for advice on who to contact with concerns relating to historic events. No details were given. The club advised he ring the NSPCC hotline in line with SFA guidance."
The club added that it regarded the safety and wellbeing of the children and young people who have contact with the club as being of "paramount importance".

Diabetes UK Cymru says there are now 177,212 people in Wales living with the condition.
It said if the trend continues these figures could rise to 288,000 by 2025.
Launching Diabetes Week, the charity said a lack of education on managing the condition had led to "devastating" health complications, including amputations, blindness and strokes.
According to the charity, many cases of type 2 diabetes can be prevented but there is no way of preventing type 1.
Diabetes costs NHS Wales nearly Â£500m a year, it added.
Diabetes UK Cymru's director Dai Williams said: "Over the last decade we have seen the number of people with diabetes rising at an alarming rate and these latest figures are a stark call to action.
"We must act now or face the very real danger of diabetes devastating the lives of even more people, and threatening to wreck the already over-burdened NHS."

About 10 hotels, pubs and club houses were also closed, officials said.
Some estimates put Lagos' population at around 20 million, creating a constant background of noise - from the blaring of car horns, to the Muslim call to prayer and loud singing in churches.
The state government has vowed to make the city, the biggest in Africa, noise-free by 2020.
Africa Live: More on this and other news stories
In August, the Lagos State Environmental Protection Agency (LEPA) closed 22 premises after residents complained about noise emanating from them.
Following the latest crackdown, its general manager Bola Shabi said the agency would no longer allow people to pray in makeshift buildings and tents.
Mr Shabi said noise levels had been reduced by about 35%, but this was not a "pass mark yet".
"Enforcement is a continuous exercise and we have set a target for ourselves. We want to ensure that Lagos is noise-free by the year 2020," he said.
Mr Shabi said mosques complied with their instructions more than churches because when they are ordered to shut down, they "instantly bring down their speakers or reduce the noise they make''.
Nigerians are extremely religious, with a large number of evangelical churches operating in Lagos.
Christians form the majority in the city.
In 2014, 116 people died when a building owned by popular televangelist TB Joshua collapsed in Lagos.

Under the bill, the performers - often dressed as popular characters like Elmo - must work for tips in special areas.
Some performers have been criticised for being overly aggressive and one was arrested for punching a police officer.
They threatened to tarnish the area's newly acquired family-friendly image.
In recent months, topless women called desnudas joined the superheroes and cartoon characters jostling for tips.
City leaders feared the desnudas in particularly would hold back their efforts to establish Times Square as a top tourist and shopping destination.
In the 1970s and 1980s, the area - filled with strip clubs - had a seedy reputation.
"People should be able to walk through Times Square without being harassed and harangued," Councilman Corey Johnson told the New York Times.
Mayor Bill de Blasio - who had considered removing Times Square's pedestrian areas altogether to combat the problem - is expected to sign the bill into law.
Partick Thistle Football Club has said it has identified a historical allegation of abuse made against a former club physiotherapist.
[NEXT_CONCEPT]
The number of people with diabetes in Wales has reached an "all-time high", a charity has warned.
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Authorities in Nigeria's Lagos State have shut 70 churches and 20 mosques in an attempt to reduce high noise levels.
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New York has passed a bill that creates rules for Times Square's costumed performers, after tourists and business interests complained that the characters were becoming a nuisance.