Flagstone has also drawn interest from Chinese private equity firm Hony Capital and could be valued at between $800m and $900m, said Reuters, citing people familiar with the matter.
The sources noted that the negotiations have not been finalised.
Earlier this year, it was reported that TreeHouse was circling egg and dairy producer Michael Foods, which was subsequently sold to Post Holdings for $2.45bn.
Earlier this year it was reported that Gryphon hired banks to manage the sale of the company.
Gryphon bought House of Nuts and Amport Foods in 2010.
The San Francisco-based firm raised $365m for a bridge fund called Gryphon Partners 3.5 in October last year, well above its $250m target.
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