The request was made at a scheduled board meeting last week although aspects of his behaviour and indiscretions about club matters had been raised at the two previous board meetings following a complaint by a supporter.
Herald Sport understands that if Murray had agreed, an amicable departure could have taken place, with the chairman remaining at the club until after the interim financial results are announced next month and perhaps even until the end of the season.
However, a second complaint about Murray from a supporter brought the matter to a head, leading to the directors requesting that he stand down at last week's board meeting.
Murray has been chairman since the consortium led by Charles Green bought the club last summer. He was introduced to Green, Imran Ahmad, the commercial director, and Brian Stockbridge, the finance director, by Zeus Capital, the Manchester-based investment bank.
He seemed a shrewd choice as chairman given his credentials as a lifelong Rangers supporter and his solid background and experience in investment and finance.
Herald Sport understands that it was a board decision to ask Murray to step down, in the best interests of the club. No deadline was set but it is unlikely the current scenario can remain indefinitely.
There have long been rumours of tension between Murray and Green, but it was concerns about Murray's private life that led to the request he stand down.
Murray has not responded to the board, although a story alleging a rift with Green and containing some details about the board meeting appeared in one newspaper yesterday.
"There are always divisions and disagreements on boards, which generally remain within the boardroom," a source told Herald Sport.
"The board supported the view that Charles' sole interest was the club, what's good for the club, and are bitterly disappointed that a difficult matter that should have been handled sensitively was exposed in such a crude and distorted manner."
Since there has not yet been an annual general meeting and no director has been formally voted into place by the shareholders, following the successful launch of the club on the Alternative Investment Market last December, the board can effectively sack the chairman.
Herald Sport understands , however, that there is a marked reluctance to pursue that course of action.