Leading South African news outlet Rapport is reporting that South African Rugby is considered Rennie to replace current coach Allister Coetzee, who has overseen the worst Springbok season in history.
Coetzee is already on record as saying he expects to be sacked after losing nine tests this year, including a first loss to Six Nations minnows Italy, ending the season with a miserable 33% winning record.
According to Rapport, South African officials have already begun work behind the scenes to secure Rennie, who is stepping down from Super Rugby franchise the Chiefs at the end of the 2017 competition.
Coetzee is set to face a performance appraisal by a sub-committee of SA Rugby, and will know after December 13 whether or not he will remain in the top job.
Rapport claimed Rennie, who led the Chiefs to the Super Rugby title in 2012 and 2013, has strong support at the highest level of SA Rugby. The news outlet calimed other coaches under consideration include former Boks mentor Jake White and Rassie Erasmus.
However, it also claimed "a potential stumbling block" to any removal of Coetzee will cost R13m ($NZ1.32m) to pay him out of the remaining two years of his contract. SA Rugby is in a dire financial situation, with the governing body reported to be as much as R19m ($NZ2m) in the red.
The other problem is Rennie signed a two-year deal with Glasgow in Scotland, with an option of a third, several months ago.
At the time, he said he had aspirations for an international role.
"There'll be a lot of international jobs come up after 2019 but ultimately the thinking is it will give us a couple of years at Glasgow and then give us a chance to see how things have gone," he said at the time.
"If we're loving it there we may be keen to stay longer. It's what has tended to happen."