The owners of United Biscuits, Blackstone and PAI, started an auction for the savoury part of the company in August.
The sale, which is being handled by Credit Suisse, is expected to raise in the region of £500m and has attracted widespread interest.
The latest from the City suggests Calbee has entered the fray, along with CapVest, Pamplona Capital and Intersnack, the maker of Vico and Penn State pretzels.
Lion Capital is understood to have shelved its interest in the business, which produces Hula Hoops and KP Nuts.
United Biscuits is not commenting on the auction – which is due to complete by the end of November.