The Ibrox club opened the door for Frank Blin, the former executive chairman of accountancy firm PricewaterhouseCoopers Scotland, to join the board in an attempt to head off calls for an Extraordinary General Meeting which have been led by engineering tycoon Jim McColl.
In a statement, the club also appeared to echo Mr McColl's desire to see people with more corporate boardroom experience at that level in the club.
The compromise offer between the rival groups came yesterday, hours after the board unanimously agreed to end former chief executive Charles Green's consultancy role.
It effectively means the group headed by Mr McColl must now decide to accept the compromise or push for an EGM. He was unavailable for comment.
They had previously called for the removal of chief executive Craig Mather, finance director Brian Stockbridge and non-executive director Bryan Smart from the board, and the appointment of former Ibrox director Paul Murray.
However, the Rangers statement fell short of those demands, saying: "The directors can state categorically that they have always been open to adding to the current size of the board and are actively seeking a new chairman, one who will bring instant and significant benefits.
"In particular it will be necessary to connect strongly with corporate Britain if the board's ambitions for Rangers are to be fulfilled.
"This board has been working tirelessly to find an intelligent solution to the request for a general meeting and all of the directors are open to sensible and reasonable additions. For instance, the board are not against Frank Blin becoming a director but do have reservations about other proposals."
It added: "The directors remain hopeful compromise can be reached and that everyone, the fans especially, will see this club emerge a much more stable and vibrant operation."
Earlier, Mr Green offered investor Sandy Easdale first refusal on his 7.7% stake in the club.
Fans gave a mixed response to Tuesday's developments at Ibrox.
Mark Dingwall, Rangers Supporters Trust spokesman, criticised the directors' move saying that the time for reform had come.
He said: "The directors' statement is very disappointing as it doesn't provide clarity or leadership. They should have taken the opportunity to address the concerns of those investors who have called for an EGM.
"Conducting a very public game of divide and conquer with your critics is totally unbecoming. We believe the board should have taken the opportunity to make meaningful reforms or gone straight to an EGM. The board has to face up to the reality that the fans want coherent answers regarding the spending of £40m."
Mr Dingwall welcomed the departure yesterday of former chief executive Charles Green.
He said: "I think Mr Green has bowed to the inevitable. His behaviour during the last fortnight has been reprehensible, in terms of undermining the manager then giving an interview to a Sunday newspaper that incurred the displeasure of the chief executive.
"There have also been revelations about his connections to Craig Whyte which have made his position untenable."
However, Drew Robertson, general secretary of the Rangers Supporters Association, said the board's attempt to calm the situation should not be taken lightly.
He said: "It looks like [the directors] are trying to come up with a compromise and I think a lot of people would agree with that.
"It would make sense for the club if they sat down and came to some sort of agreement that suits both the people who want an EGM, and the board.
"Some people say that an EGM is necessary to get to the bottom of what's been going on, but if we can resolve things without that then so much the better."
He added: "I do not know much about Frank Blin. Neither camp has put up a very convincing case about why they are the right people to run the club.
"The McColl camp believes there is enough concern there to force an EGM, and it does not think the board has a strong enough case to take the club forward. But we don't know enough about them either."