Robert Sarver has increased his offer to buy Rangers to £20m, which is £2m more than his bid that was on Tuesday.
Part of the offer by is a short-term loan of £6.5m to alleviate the club's cash shortage.
He said it would help the board deal with its "cash flow crunch" and address his desire to see Rangers on "a solid long-term financial footing".
Rangers say they are examining Sarver's updated proposal.
Previously, the club said Sarver's £18m bid for a majority shareholding undervalued the club.
A statement issued on Thursday said: "The directors thank Mr Sarver for his continuing interest in Rangers. They are studying the revised proposal which was received last night, and are consulting with major shareholders, to understand whether the proposal is capable of being implemented."
The 53-year-old owner of the NBA Phoenix Suns basketball franchise says he is now prepared to increase his offer to buy out existing shareholders at 20p per share rather than the original offer of 18p.
Sarver suggests that the loan offer of £6.5m would leave Rangers free of debt, allowing them to repay the loans made by shareholders Mike Ashley and Sandy Easdale.
It has since emerged that Easdale, a shareholder and the chairman of the football board, has been repaid the £500,000 emergency interest-free loan he recently made to the club.
The terms of Sarver's loan offer have not been revealed, but the Rangers statement says it must be repaid within three months.
Rangers said: "The revised proposal involves investing up to £20m for a majority shareholding by way of a placing of new ordinary shares in Rangers at 20 pence per share ("Placing") followed, if the Placing is completed, by a mandatory offer for the remaining issued and to be issued ordinary share capital under Rule 9 of the City Code on Takeovers and Mergers ("Code") at 20 pence per share.
"As part of the revised proposal £6.5m would be made available to Rangers in immediate short-term funding by way of a secured loan to be repaid in 90 days or out of the proceeds of the Placing."
|Spokesman for Robert Sarver|
|"To be very clear, Mr Sarver has never met or spoken to Mike Ashley, Sandy Easdale or James Easdale. "And for the complete avoidance of doubt, he has never met or spoken to Charles Green or Craig Whyte. He is not working with, acting for, or alongside any of these individuals. "Mr Sarver is acting on his own behalf, although he has repeatedly said that if he succeeds in acquiring a majority stake in Rangers he would like to work with others who clearly have the best long-term interests of Rangers at heart."|
The Arizona-based businessman said in a statement: "If the board agrees to my proposal, I would provide a total of £16.4-£20m to be invested in the club as the board determines - £6.5m immediately and the remainder following a share issue, taking the major step necessary to restore financial stability to the club.
"I know, of course, that there are various groups of prominent fans who have been working hard for a long time to bring change to the club and they can rest assured that I am committed to building a strong partnership with fans and key stakeholders to do what is best for the club.
"I fully realise that a strong sports club needs local investment, local directors and local support to be successful.
"I've received messages of support from many Rangers fans and really appreciate their good wishes - even if sometimes I don't immediately get their Scottish sense of humour!
"After everything they have been through in recent years, I understand some are rightly cautious about a guy they've never heard of wanting to get involved in their club. I'd feel exactly the same way.
"But hopefully I can combine my financial resources and knowledge of professional sport with strong local leaders and top-notch management to bring Rangers back to elite level."
Sarver has owned the Phoenix Suns since 2004 and is chief executive officer of the Western Alliance Bancorporation.
He is reported to have been alerted to the situation at Ibrox by former Rangers left-back David Robertson, who now coaches his three sons.
He wants a controlling stake in the Scottish Championship club but, for that to happen, the board would need 75% shareholder approval to issue that many shares solely to a non-shareholder.
Rangers' options to extricate themselves from the financial turmoil are limited.
Newcastle United owner Mike Ashley has 8.92% of the shares but his bid for almost 30% of the stock was rejected by the Scottish Football Association because it would break the agreement the Sports Direct owner had struck with them over dual ownership.
Further, two other major players, Dave King, who owns 15% of the shares, and a group comprising wealthy Rangers fans Douglas Park, George Taylor and George Letham, who own about 20%, would seek boardroom control for further major investment.
On Tuesday, former Rangers chairman Alastair Johnston questioned why Sarver would wish to invest in the club, but added: "Of all the pretenders that have come along in the past few years, whether or not he has the understanding of the emotional interest in the club that some of the others have, he has certainly got the business acumen and the financial wherewithal."
Meanwhile, The Rangers Supporters Trust say they have purchased a further 100,000 shares in the club, taking their shareholding to over 1%.