White, who led the Springboks to World Cup glory in France last month, informed SA Rugby he did not intend to extend his spell in the role, leading the governing body to advertise his position.
And, though White later hinted at a change of heart, SA Rugby opted to stick with those who had initially applied for the post.
The addition of Muir to the shortlist appears to make a mockery of that policy, but the move has nevertheless been welcomed by White, who himself was not on the original shortlist to succeed Rudolph Straeuli four years ago.
"If there had not been flexibility back then I would never have got the opportunity to coach the Boks - never mind be part of a World Cup winning squad," White told the Mercury newspaper.
"I think it is exciting that Dick - as a former Springbok and a Super 14 coach - wants to coach the Springboks.
"He understands the ethos of the Boks, he has been through the system and if he was the successful applicant there would be continuity for the squad."
The SA Rugby president's council will meet on Thursday to discuss the shortlisted candidates, with Bulls coach Heyneke Meyer and South Africa assistant coach Alistair Coetzee among the front-runners to replace White.