Rangers, which last week took an emergency loan from its chairman and has said in order to stay afloat it will need further funding beyond January, had rejected Sarver's offer because it did not think the required 75 percent of shareholders would back it.
Sarver, owner of the Phoenix Suns basketball team, had raised his offer to 20 million pounds for a controlling stake from an original offer of 18 million.
The offer, which would have required the approval of investors at a general meeting, also included a short-term loan of 6.5 million pounds to alleviate the club's cash problems.
The 54-times Scottish champions, who have climbed back to the second tier after being wound up and re-formed as a fourth-tier club in 2012, had rejected Sarver's first approach for similar reasons.
In reaction to Rangers' rejection and the club's failure to enter into any meaningful negotiations, Sarver said he did not intend to make an offer for the club.
"Clearly the current directors have a different path they want to go down. I hope for the sake of the club and the fans, who've been very warm and supportive towards me, that it works out," Sarver said in a statement.
(Reporting by Kate Holton and Neil Maidment; Editing by Paul Sandle)
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