The Springboks won the World Cup with a 15-6 win over England in Paris earlier this month and it has yet to be revealed whether White will stay in position after his contract expires at the end of the year.
He has already made noises about taking up a similar role with England if current coach Brian Ashton leaves, but Allan insists he should continue his work with the world champions.
Allan represented South Africa in 13 Tests between 1993 and 1996, having played nine times for Scotland in the early between 1990 and 1992.
He is currently the CEO of the South African Rugby Legends, a group of ex-international players.
Allan said that beating the Lions when they visit South Africa in 2009 would be a worthy reason for White to stay on as coach.
Allan said: "The only thing that will keep him on, I believe, will be the Lions series. That's a big feat - coaches have won the World Cup before, but they've never won the World Cup and had a series win over the Lions, so it's a question of his ego, whether his ego wants to stay and coach against the Lions."
White's contract expires in December and Allan said South Africa's rugby officials were right to tell him to re-apply if he wants to keep his job.
He said: "I don't think they would have gone through this process if Jake hadn't indicated that he might be leaving after the World Cup.
"I think it's a foregone conclusion that if he wants the job, he'll get it. It's whether he wants the job or not.
"Director of rugby is another position that White has been linked with - I'm not sure if that's still on the cards, but potentially that might be quite interesting.
"He was fairly positive about not coaching after (the tournament) because of all the political interference in South Africa, but now after winning the World Cup, he wants to re-evaluate and he doesn't want to rush into things," said Allan.
"Obviously there will be options all over the world now after winning coach of the year, so he must discuss it with his family."