Miller, who heads a tow-truck company in Tennessee, plans an 11.2 million-pound bid for Rangers which would create what he described as an "incubator" company while the administrators reached agreement with the club's creditors, local media reported.
Miller said he would suspend his offer until Monday to allow other bidders to "put up or shut up".
Singapore businessman Bill Ng withdrew from the race earlier on Friday in the latest blow to administrators seeking to rescue the debt-laden Glasgow club.
In a statement carried on the Straits Times website, Ng said his five-man consortium had become "increasingly frustrated with the process of dealing with the club administrators".
However, the report added that he could relaunch his bid if administrators Duff and Phelps failed to reach a deal with other potential bidders.
Rangers, Scottish champions a record 54 times, went into administration in February over unpaid tax bills. They suffered a 10-point penalty and have recently lost their league title to Glasgow rivals Celtic.
The Blue Knights consortium led by former Rangers director Paul Murray withdrew from the bidding this week. However, reports have said they would be ready to revive their interest.
Brian Kennedy, a Scottish businessman who owns English rugby club Sale Sharks, has made a verbal offer.
Miller portrayed his proposal of a way of preserving the 140-year-old club rather than it having to be liquidated and potentially restart life at the bottom of the Scottish game.
There are a number of issues complicating efforts to rescue Rangers. The club owes around nine million pounds in unpaid taxes and faces a much larger potential liability in another tax dispute dating back over a decade.
Craig Whyte, who bought an 85 percent stake in the club a year ago for a nominal one pound, also raised more than 20 million pounds against future season ticket sales in a deal with a company called Ticketus.
The Scottish Premier League is also meeting at the end of the month to decide if clubs which go insolvent and are then reformed should be docked points in future seasons.
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