Ibrox chairman Dave King made the call following reports over MacLennan’s recent appointment as non-executive chairman of Irish-based Independent News & Media Group (INM).
Celtic’s largest shareholder, Dermont Desmond, is listed as a major shareholder in INM along with Denis O’Brien, who has a minor stake in Rangers’ Glasgow rivals.
The SPFL has insisted that MacLennan informed its board of his new role before taking up the offer and denied claims there was a conflict of interest.
MacLennan joined the board of INM, which publishes titles including the Irish Independent and Belfast Telegraph, on March 1.
In a statement on the Rangers website, King said: “The club notes with concern the latest disclosure through the media regarding a business relationship which the chairman of the SPFL has with leading shareholders of a fellow SPFL club.
“This has given rise to allegations of non-disclosure and it is now imperative that we discover exactly the nature of the information supplied to the SPFL regarding this relationship.
“It is equally important that any conflicts of interest, or even the perception of such, whereby positions within the Scottish football authorities could be undermined, or abused, must be aggressively rooted out of our game.
“The SPFL, therefore, must immediately suspend its chairman pending an independent investigation by a senior QC into the allegations and into the extent that other parties within the SPFL may have co-operated in this alleged non-disclosure.”
In response, a spokesman for the SPFL said: “It’s not surprising that Murdoch, having stepped down as deputy chairman of Telegraph Media Group, should be approached by other businesses in that sector.
“The members of the SPFL board were each informed of Murdoch’s appointment on January 19 2018, the same day it was publicly announced in a press release from Independent News & Media PLC. That release was also circulated to the entire SPFL board that day.
“The appointment was widely reported in the national media at the time, including in the Sunday Times, Belfast Telegraph and Sunday Independent. No director raised this issue subsequently and it was not the subject of any board discussions.
“To be definitive, a non-executive position on a PLC does not constitute a business relationship between that individual and a minority shareholder in the company and therefore no investigation is warranted.”
Rangers’ intervention marks the continuation of an increasingly fractious relationship with the Scottish football authorities.
Last week King called for Scottish Football Association board member Gary Hughes to be suspended over claims he described Rangers fans as the “great unwashed” in an article in a trade magazine in 2006.
Rangers were hit with two SFA disciplinary charges the previous week relating to financial information which allowed the Ibrox team to play European football in 2011-12.
In his latest statement, King added: “The credibility of Scottish football in the minds of supporters and sponsors is at stake and urgent action is required.
“The SPFL and SFA must now appoint independent investigators. Scottish football is an important national asset and must have levels of probity and governance that are beyond reproach and that are transparently so.”