Roux quits senior South African rugby post
Media reports say Rathbone, boss of the Pretoria-based Bulls, may be slight favourite to succeed Roux, who has been with the South African Rugby Union since 2013.
Rathbone is a former junior-level scrum-half while Sharks chief Coetzee was a prop with franchises in South Africa and clubs in France.
The departure of Roux had been predicted for some time after an audit firm said he illegally moved 37 million rand ($2.1mn/2.0mn euros) at Stellenbosch University to their rugby club.
He lost an appeal against the finding and has been ordered to repay the money to his former employer.
South African rugby boss Mark Alexander said "Roux had resigned in the best interests of the game and the organisation.
"Jurie agreed to step down at the end of December. It is sad and he resigned due to things beyond his and our control," added Alexander.
South Africa are the reigning Rugby World Cup champions and Roux oversaw the switch of the major domestic franchises two years ago from southern to northern hemisphere competitions.