Legal tussles are threatening to drill a hole in a New York City celebrity dentist’s business plans.

Dr. Jonathan B. Levine — who has appeared on “The Dr. Oz Show” and counts Christie Brinkley among his clients — is mired in state and federal court battles after a company he co-founded eight years ago accused him of infringing on its trademarks for a line of teeth-whitening products.

Go Smile — whose products include a light-powered teeth-whitening device for home use called the “Smile Whitening System” that began selling on the QVC shopping channel in August — claims that Levine plans to launch a similar device as soon as next month called the “Glo Smile Whitening System.”

That would be a “confusingly similar” name for the new device, Go Smile charged in its lawsuit. The company noted that Levine sold his interest in Go Smile years ago and cut all ties with the company.

However, a lawyer for Levine contends that no such name is planned for any product coming from his client. The lawyer, David Denenberg, also noted that a state court this fall blocked Go Smile from improperly using Levine’s name to promote its own products.

Go Smile, as a piece of evidence attached to a lawsuit filed last month, submitted a Web page from Levine’s site that, in one instance, uses the name “Glo Smile Whitening System” to describe the new device. As of yesterday, however, the name no longer appeared on the page, which instead called the device the “Glo Brilliant Smile Whitening System.”

Denenberg said the original phrase may have been a typo. The legal eagle said it hadn’t been changed because of Go Smile’s lawsuit. “It’s not being used now for any product, so there’s no issue,” Denenberg said.

Likewise, the lawyer charged that it was “rather ironic” that Go Smile had made such accusations, given that it has already been slapped for improperly using Levine’s name.

Sure enough, a New York judge in October ordered Go Smile to pony up for long-unpaid consulting fees to Levine, even as the judge rejected Go Smile’s demand for access to the tooth doctor’s “trade secrets” for making and marketing new teeth-whitening products.

More broadly, Go Smile charges that Levine is rolling out a “Glo-based family of trademarks” that are “confusingly similar” to its own “Go-based family of trademarks” — an allegation disputed by Levine. james.covert@nypost.com