Details of payments to former managers, players and other staff were featured in the documentary Rangers – The Men Who Sold The Jerseys and on a number of websites, including the anonymous Rangers Tax Case blog.
The businessman's call came after he expressed concern about the level of information linked to the HMRC tax tribunal that was made public prior to this week's ruling in the oldco club's favour.
More than 12,000 people had by last night signed an e-petition calling on the UK Government to launch a probe into HMRC's investigation and the suspected leaks.
Sir David's Murray International Holdings (MIH), which was the majority shareholder in Rangers until Craig Whyte bought the club in May 2011, said in a statement that individuals had an expectation of privacy when it came to correspondence and matters relating to financial affairs.
The BBC named a number of people who received payments under the controversial Employee Benefit Trust, the subject of the HMRC investigation, and named Sir David as the chief beneficiary of the scheme following a tax-free payment of £6.3 million.
The income of a number of players and former club figures were also made public by the broadcaster.
An MIH spokesman said: "It is disgraceful that personal information relating to employees and former employees of MIH and its subsidiaries has been bandied about in public in such a casual manner.
"There were only three potential sources of the complete set of documentation utilised in the broadcast and correspondence: MIH's head office, the First Tier Tax Tribunal and HMRC, together with their respective advisers. Importantly, Rangers Football Club plc did not have access to all of the material covered in the programme and letters.
"MIH's underlying concern is that there may have been criminal offences committed in connection with the provision of this material to the BBC. MIH therefore requests a formal independent police inquiry into how this documentation came into the possession of the BBC.
"In this regard, MIH will willingly and openly co-operate with any formal investigation."
Strathclyde Police said no formal complaint had been received. BBC Scotland declined to comment. Rangers – The Men Who Sold The Jerseys won the current affairs category at the Scottish Baftas.
The petition, which will require 100,000 signatures to be debated in Parliament, said: "Throughout this 'investigation' there have been several leaks of confidential information relating directly to sensitive information about the club, the employees and the current state of play within the 'investigation'.
"The source of this leak must be identified and dealt with accordingly due to the serious breach of protocols."
The oldco's liquidator, BDO, confirmed it would pursue an investigation into issues at Ibrox that led to the HMRC investigation and to the financial collapse at Ibrox.
Former Ibrox director Paul Murray reportedly claimed BDO had the power to forensically examine the activities of all involved with the club in the run-up to its failure, including Craig Whyte and HMRC, in a bid to recoup some of the millions owed to creditors.
BDO said: "By investigating the reasons for the company's failure, we will better understand the avenues available to enable the recovery of all possible monies for creditors. This is a complex case with many potential areas for us to investigate."
Meanwhile, Rangers manager Ally McCoist said: "I have always been of the opinion – and I don't care what anybody says – that any mistakes David Murray made were made in trying to do the best for the club."
l A 37-year-old man is due to appear at Glasgow Sheriff Court next month after he allegedly posted offensive material on the Rangers Tax Case blog. The accused went voluntarily to a police station in the city where he was arrested by officers from the national Football Co-ordination Unit, after the force was alerted to the material.