The Scottish-born businessman last week reached an agreement with the South African Revenue Services over a lengthy tax feud in his adopted homeland.
King, who ploughed £20million into Rangers during Sir David Murray's reign at Ibrox, has paid £45million to SARS to settle his tax issues and could make a move to invest more in the Light Blues.
Greenock businessman Sandy Easdale, whose brother James is a Gers director, confirmed last week he had finalised a deal to purchase former chief executive Charles Green's majority shareholding in the club while a battle for control at Ibrox is ongoing, spearheaded by former Blue Knights leader Paul Murray and Scottish businessman Jim McColl.
King has revealed in recent weeks that he has been offered shares in Rangers but described the asking price as "absurd" and he is determined not to hand over money to current shareholders as he looks to once again help his boyhood heroes.
King said: "It is something that is on my mind. I have had discussions, at an early stage, over the last couple of months where it looks like the club will need some more funds.
"At this point in time, I would be interested in putting additional funds into the club by way of, say, a rights issue, but what I am not keen to do is to do an acquisition of existing shares where the money goes to the shareholder, not into the club."
King has long been seen as a potential saviour of Rangers by supporters, many of whom are backing calls for sweeping changes to be made in the Ibrox boardroom.
Murray and McColl put forward a requisition for an EGM to be held in a bid to oust chief executive Craig Mather, finance director Brian Stockbridge and director Bryan Smart from the board.
King could now be ready to invest more money into the club after bringing an end to his tax battle.
He said: "When this whole thing started in 2002, I adopted a bit of, let's say, a siege mentality.
"I thought it was an onslaught, there were criminal charges, there were fraud allegations, there were exchange control issues. I really took this total onslaught approach.
"I was genuinely, very pleasantly surprised that when I did engage with SARS, this didn't happen over a couple of weeks, this has been going on probably for a year of serious negotiations.
"Make no mistake, SARS were very firm in the negotiations, they were very tough in the negotiations but I did find they were fair, they were open and, if I look back, it is something that I regret that I didn't do sooner."