The New York Times is reporting that OPI Products, the popular nail polish company known for its quirky color names, is up for bid. The newspaper reports that according to Bloomberg News, OPI is in advanced talks to sell itself to a beauty care company or private equity firm for as much as $1 billion, citing four people with knowledge of the matter.

Coty, the closely held perfume maker, is thought to be the front-runner, while Bain Capital and Advent International, the private equity firms, are also among those that made first- round bids, according to the report. Second-round bids are due in the next two weeks, Bloomberg reported.

The company, based in Hollywood, has tapped Lazard to run the auction, which could yield between $900 million and $1 billion, the news service said.

OPI’s nail polish, a standard at many manicurists, has gained a reputation for its clever, hard-to-copy color names. Its fall 2010 collection, which takes its inspiration from Switzerland, includes shades like “William Tell Me About OPI” and “Lucerne-tainly Look Marvelous.”