The investors want former Ibrox director Paul Murray, Sandy Easdale - brother of current non-executive director James - and accountancy expert Frank Blin appointed as directors.
The group have also proposed the return of ex-chairman John McClelland to the board.
The proposal was put to the club on September 4, with the shareholders signalling, if accepted, they were ready to withdraw a request for the removal of chief executive Craig Mather and other key directors.
Rangers are keen to avoid an EGM but have so far failed to reach agreement that could see proposed boardroom changes folded into the business of the club's annual general meeting which is set to take place in October.
A statement to the London Stock Exchange on Tuesday morning confirmed: "Further to the announcement on 4 September 2013, the Company confirms that whilst the Board's discussions have continued with representatives of the group who requisitioned a general meeting to consider the proposed resolutions detailed in the announcement on 2 August 2013, the Board is disappointed to announce that no agreement has been reached.
"Unless an agreement can be reached prior to 13 September 2013, as set out in the announcement on 2 September 2013, the Company will be required to send a notice to shareholders to convene the General Meeting by no later than 13 September 2013.
"Further announcements will be made as appropriate."