So far, most of the reports involve Verizon Wireless users in the U.S. But there are a few reports on tech blogs and media sites that users on other networks have been affected as well.
"In six days, I've used over 6 gigs of data. I've been on Wi-Fi over 80 per cent of the time. This is crazy!" wrote one user, who reported his data surge last week on an Apple discussion board. "There is definitely a problem here. I used less than a gig monthly when I was on the iPhone 4," he wrote.
The technology site CNET says it received an email from a Canadian user who was complaining of a similar problem.
Representatives with Bell Mobility, Rogers Communications and Telus all said they have heard of no problems among their users. "The data bleed issues reported in the U.S. do not apply to our OS version and customers should not be experiencing the same issues here," according to emailed comments from Rogers. Apple has not yet responded to a request for comment.
Apple released a software fix on the weekend to address the bug for Verizon Wireless customers. Apple hasn't formally commented on the extent of the problem, except to acknowledge in a brief note on its support site that the patch was designed to resolve an issue in which the "iPhone 5 may use Verizon cellular data while the phone is connected to a Wi-Fi network."
Verizon reassured its customers that they wouldn't be charged for any unwarranted data overages.
Some users are speculating that when the phones encounter difficulties connecting with Wi-Fi networks, the phones switch over to the fast new LTE networks, racking up unexpected cell data charges. Others wonder if the phones are incorrectly counting free Wi-Fi data as billable cellular data.
Some say it appears to be a problem with the new iOS 6 operating system, reporting that older iPhone models that have been upgraded with the new operating system are now chewing through data.
The data glitch isn't the only headache Apple has faced with its popular new iPhone 5s. Apple CEO Tim Cook apologized last week for the release of Apple's new Maps application, which comes with the new phones. The app received a barrage of criticism for its shortcomings.
Apple sold more than five million iPhone 5s during their first weekend of wide release last month.