SFA rules state that no individual can have a dual interest in two clubs, although Sports Direct kingpin Ashley's MASH Holdings Ltd already holds an 8.92 percent stake in Rangers.
"The Board has now carefully considered the application and has decided, unanimously, that the application should not be granted," an SFA statement said on Wednesday.
"The Board, under Article 13 of the Scottish FA Articles of Association, is required to have due regard to the need to promote and safeguard the interests and public profile of association football, its players, spectators and others involved with the game."
In October Ashley provided two million pounds of funding for the financially-stricken club in return for the removal of chief executive Graham Wallace.
Ashley has said he has no intention of selling English Premier league club Newcastle for at least two years, but his motives have been called into question by the SFA who have asked for clarification from Rangers about his intentions.
"Except with the prior written consent of the board no person who is involved in any capacity... in the management or administration of a club may at the same time either directly or indirectly be involved in any capacity... in the management or administration of another club," the SFA said this month.
Rangers were demoted to the fourth tier of Scottish football in 2012 due to financial difficulties.
They have earned consecutive promotions but this week manager Ally McCoist left his post after a run of poor results.
They are currently second in the Championship with 35 points from 17 games, 12 behind leaders hearts.
(Reporting by Martyn Herman; Editing by Ken Ferris)
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