It is understood that cash disappeared from a bank account run by the National Museums of Scotland (NMS).
According to reports yesterday, two members of staff from the finance department were suspended after the discovery of the fraud which dates back several years.
As a result of the internal investigation one member of staff was dismissed and the other one resigned.
The matter is currently the subject of a police investigation.
The discovery that more than £80,000 was missing was made after an investigation by accountancy firm KPMG.
An NMS spokesman said: “We became aware of the issue in July 2012 and started an investigation immediately with the involvement of our internal auditors KPMG.
“Since then it has been confirmed that there has been a fraud. The matter has been passed onto Lothian and Borders Police.
“Two members of staff were suspended and following internal disciplinary procedures, one member of staff was dismissed and the other resigned.”
According to NMS accounts for 2011-2012, auditors were “engaged to review the integrity of certain cash withdrawals following discrepancies which came to light during July 2012”.
It was on the basis of this report that museum bosses decided to call in the police.
A police spokesman said: “Lothian and Borders Police can confirm that we are investigating a report of historic fraud at the National Museums of Scotland.”
Scottish Liberal Democrat leader Willie Rennie said: “The NMS does fantastic work with schools, students and the wider public to educate them about the history of Scotland.
“However, it is also a body in receipt of a substantial amount of public funding. The allegations of historic fraud that have been reported are clearly very serious.
“I know that management at NMS will be anxious to get to the bottom of this and it is right that the legal process is allowed to run its course.
“When it comes to bodies and organisations in receipt of public support, ministers need to be aware of exactly how taxpayers’ money is being spent.
“The Scottish Government will need to follow this case closely and ensure that any lessons on audit processes and accountability are taken fully on board.”
Sarah Boyack, Scottish Labour MSP for Lothian, said: “I am deeply concerned by the allegations and I welcome the police investigation. It is important that strong action is taken if it is revealed that public money has been misappropriated.”
A Scottish Government spokesman said: “The National Museums of Scotland has kept the Scottish Government informed about the case.
“This case is currently being investigated by Lothian and Borders Police, therefore we are unable to comment further at this time.”
NMS receives more than £20 million a year from public funds.
The body operates the National Museum of Scotland in Chambers Street, Edinburgh, as well as attractions, including the Museum of Flight in East Fortune and the Museum of Rural Life in East Kilbride, Lanarkshire.
The Chambers Street museum reopened in 2011 after a £47.7m refurbishment.