The Lions forward was injured in last November's autumn Test series and is working his way back to full fitness ahead of next month's tournament.
White insists he is happy with the progress he is making following the serious knee injury but is well aware he will not be at his inspirational best for the Pool C opener against Portugal.
Stiffer challenges follow against favourites New Zealand and especially the vital game against Italy, who Scotland face in the last game in the pool with a place in the quarter-finals likely to be at stake.
After nine months on the sidelines, the big-hitting back-rower admits it is taking him time to regain his match-sharpness with the crunch with the much-vaunted Azzurri forwards in St Etienne looming large.
White said: "I was delighted to come through the game against South Africa although I'm sure like all of the boys I will be feeling the effects of the game at the start of the week.
"The intensity was a notch up from the Ireland game and personally I'm very disappointed that we missed a chance to push the Boks close.
"That didn't happen but we're going to take stock and try to improve week on week because the main thing is to qualify for the quarter-finals.
"My own sharpness is getting better but I'm realistic and know it will take a couple of games to feel that I'm playing naturally. However, I'm pretty happy with my progress.
"We're not used to playing at the intensity that the South Africans bring and the more we play together and the more we play international rugby against sides like these we will get better.
"The fact they have had more games than us was a definite factor and the longer we have together at the World Cup the better we will become."