Former captain and manager Greig and businessman McClelland, who has been with the club 11 years, submitted their resignations to chairman Craig Whyte. The pair asserted that their roles have been sidelined since Whyte assumed ownership in May.
A statement to the PA said: "[The resignations] result from John McClelland and John Greig both being of the opinion that since the change of ownership they have been excluded from participating in corporate governance at the club.
"Therefore they have decided not to continue in their positions as non-executive directors."
Rangers currently sit top of the SPL, seven points clear of second-place Motherwell.