Ashley has courted regular controversy in his seven-year stint in charge of Newcastle and was said to be considering saving Rangers from further financial issues.
The 50-year-old is listed as having a 4.6 per cent stake in Rangers while it was recently revealed that he owns the naming rights for Ibrox Stadium.
Ashley released a brief statement on Friday which said: "I will not be participating in the open offer which closes at 11am today, Friday, September 12, 2014."
Rangers' latest share issue is hoped to raise approximately £4 million, with the results set to be revealed on Friday.
The Scottish club play in the second tier but are likely to return to the top flight next season.
UEFA regulations state that the same person cannot own two clubs that could play each other in European competition.
And while that is not an issue at the moment, Ashley is reported to be keen on selling Newcastle – although it appears a move to strengthen his position at Rangers is now dead in the water.
