Sources in the city revealed that Mather had flown out to try to convince King to make a return to the club in an influential role.
King has long maintained that the current regime at Ibrox will need further investment due to the business plan put in place by Charles Green, the former chief executive, being unlikely to return Rangers to the top flight with enough resources to immediately challenge for the title. Having been unable to bid for the club last summer due to issues with the South African tax authorities - which have now been resolved - King is able to freely invest at home and abroad, and has been biding his time.
The businessman, who previously invested £20m in Rangers under Sir David Murray, is prepared to wait for the right moment before making his move to buy either a controlling or ownership stake in the club, having identified the eventual return to the top flight as the moment when fresh investment would be crucial.
Currently, Sandy and James Easdale, the owners of McGills buses, either own or have proxy control over almost 25% of the shares in Rangers International Football Club, but Mather is thought to believe that King would be a better long-term option for the club.
King would be unlikely to contemplate returning unless it is to a position on the board and Mather is understood to consider that the talks went well.
King may believe that it would be more prudent to wait than accept any offer to return to Ibrox now, though, with supporters' protests against the board growing in intensity.
A battle is looming at the annual general meeting on October 24, with a group of shareholders pushing for the current board of directors - Mather, the finance director Brian Stockbridge, and the non-executives Ian Hart, Bryan Smart and James Easdale - to be voted off the board and replaced by the former Rangers director Paul Murray, the former Ibrox chairman Malcolm Murray, and two Rangers- supporting businessmen, Scott Murdoch and Alex Wilson. The Ibrox club is also without a chairman since Walter Smith stood down last August.
The current board has sought to block the director nominations, though, and a hearing at the Court of Session on Monday will rule on whether or not the nominations of four replacement directors should be added to the order of business of the AGM.
This could, in turn, lead to the AGM being delayed to allow shareholders enough time for due consideration.