White's contract expires on December 31 and SA Rugby will conduct interviews with four candidates in January to appoint the World Cup winner's replacement.
Coetzee is on a shortlist along with Super 14-winning coach Heyneke Meyer, former Cats coach Chester Williams and IRB World Under-21 championship-winning coach Peter de Villiers.
White believes it would make sense for SA Rugby to appoint Coetzee to the position.
"For the sake of continuity that would be the best decision," he said. "SA Rugby has invested a lot of money in Allister's development as a coach in the past decade.
"He's already coached the Under-23s and the SA 'A' team, and the time has now arrived to give him the senior post.
"What I would suggest is that Allister should be surrounded by a strong coaching team. People like Heyneke Meyer, Dick Muir and Rassie Erasmus could add a lot to Springbok rugby.
"It only makes sense to have as many experienced coaches as possible at the helm. Just look at New Zealand. In Graham Henry, Steve Hansen and Wayne Smith they had three international coaches leading the team.
"I know they lost in the World Cup tournament, but a record of 42 wins in 47 Tests is certainly nothing to be sneezed at either."
With his departure from the Springboks imminent, White revealed that he does not intend to coach in South Africa again.
"I've won the Under-21 world title and the senior World Cup. I think I've achieved what I could here and won't be able to coach at a lower level again," he added.
"I'm not considering any overseas offers at the moment either, but nor am I saying that I'll never coach again."
White also believes that it would not be a betrayal to coach another side against the Springboks next year.
"Look at Nick Mallett. He's now coaching Italy and they are due to play here next year. And Eddie Jones didn't think he would ever help coach South Africa.
"But it's the age of professionalism and these things will happen increasingly," concluded White.