The Rangers Supporters' Trust has also called on Green and former club director Imran Ahmad to take legal action against former owner Craig Whyte following claims he was involved in the Sevco deal that saw Green assume power at Ibrox.
Green, 59, has been at the centre of a media storm in the last week.
He has been issued with a Scottish Football Association Notice of Complaint, accusing him of breaching two rules in relation to "offensive and racist comments" made about Ahmad in a newspaper interview, for which he was also criticised by campaign group Show Racism the Red Card.
Rangers fans were also angered by reports suggesting he had instructed manager Ally McCoist to dispense with the services of right-hand men Kenny McDowall and Ian Durrant.
An RST statement said: "The board of the Rangers Supporters' Trust today calls for the future of chief executive Charles Green to be decided by the board of the club and for Charles Green and Imran Ahmad to pursue legal action against Craig Whyte.
"Our club needs clarity over two major issues - the future of Charles Green and the ongoing claims of Craig Whyte. The proper places for these two to be handled are, respectively, the Ibrox boardroom and a court of law.
"The board of the club has enjoyed the backing of the overwhelming majority of the Rangers family and to date no valid concern about its operation or composition has been aired.
"Therefore, it has the competency and authority to deal with the matters surrounding Mr Green's recent performance without interference from the media or external bodies."
Whyte says that he still has a claim to the club's assets and that he was behind the consortium, headed by Green, which took over the club, with Green merely acting as a front man.
Green then transferred the assets - which were bought for £5.5m in June last year - to a different company called Sevco Scotland, which then became The Rangers Football Club.
But Whyte is apparently arguing that such a transfer is illegal and has threatened legal action against Green and Ahmad.
The RST statement continued: "Craig Whyte will never be an acceptable owner or investor in Rangers - his allegations appear so outrageous and injurious to the club that only a court appearance will force him to provide conclusive evidence of his claims.
"As he is alleging collusion between himself, Mr Green and Mr Ahmad we believe that the situation now requires judicial examination and recommend that Mr Green and Mr Ahmad instigate legal proceedings against Craig Whyte without delay."