White, 33, broke down while bowling in the fourth Test against Australia in Melbourne on Friday, but batted on Saturday, scoring 85 not out in England's first innings of 270.
"I had a scan last night and it showed I have a tear in my side muscle. It could be six weeks before I'm right to bowl again," White told a news conference on Saturday.
"For me personally, the World Cup (starting in February) is my main aim. To try and get fit for that, that's six weeks away and this injury is normally six to seven weeks so it's probably going to be touch and go."
White is unlikely to bowl if Australia bat a second time in the fourth Test. England are 111 for two in their second innings after three day's play and require another 170 to make Australia bat again.
White added if he was unable to bowl in next week's fifth Test in Sydney, which starts on January 2, he was unlikely to play.
"It was sore when I was batting. Cross-bat shots were quite painful. If I'm not bowling in Sydney, I don't think I'll be playing," he said.
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