The Rangers Supporters' Trust (RST) has also called on Green and former club director Imran Ahmad to take legal action against former Rangers owner Craig Whyte following claims he was involved in the Sevco deal that saw Green assume power.
Green's outlandish comments in recent days have put him at the centre of a media storm. On Wednesday, he was handed a Scottish Football Association Notice of Complaint after referring to Ahmad as his "little P*** friend" in a newspaper interview.
He has also caused upset amongst fans after reports emerged suggesting he had instructed manager Ally McCoist to dispense with the services of right-hand men Kenny McDowall and Ian Durrant. Now the RST has called on those within the Ibrox boardroom to make a decision about his future based on his current performance as chief executive.
A 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 claims he was the main driver behind the Sevco 5088 company which purchased the assets and business of the company he had put into administration. Days later Green transferred the assets - which were bought for £5.5million in June last year - to a different company called Sevco Scotland, which then became The Rangers Football Club. Whyte's main argument appears to be that such a transfer was illegal and he remains the rightful owner of the club's assets.
The RST statement continued: "With regard to Mr Whyte's claims, the Board can question the directors and employees concerned in a confidential manner which should provide additional clarity. 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."