Commenting on bristolshoguns.co.uk White said: "It's the same injury as the other week and it's flared up again. I was hoping to be involved on Saturday."
The Shoguns and England prop picked up the injury in the Zurich Premiership fixture against Leeds at the beginning of February and missed the Shoguns' last fixture against Leicester.
The front row expert made a good recovery and played an impressive 80 minutes for the England side against the French in the RBS Six Nations opener and had hoped to continue against the Welsh.
"I'm very frustrated to be out. Saturday was another opportunity for me and the injury has ruled me out," added White.
Shoguns head coach Peter Thorburn said: "We're disappointed for Julian as he played so well against the French last Saturday."
The 29-year-old tighthead prop is set to have an operation, by surgeon and former Bristol and England fullback Jonathan Webb, on Thursday afternoon. The operation will repair knee cartilage damage.
"Julian has got himself in such good physical condition it's best for him to have the operation as soon as possible. Jonathan Webb has been superb and fitted Julian in on Thursday at very short notice," added Thorburn.
White is expected to make a full recovery within the next three to six weeks and will be targeting England's final games of this year's RBS Six Nations as well as the Shoguns fixture in early April for a return.