Berry, who has a 4.1 percent stake in the company, has reached out to several buyout firms, seeking a private equity partner in order to put together an offer for Fresh Market, the people said this week.

Private equity firm Apollo Global Management LLC has already agreed to work with Berry on the potential offer for Fresh Market, the people said.

Berry's son-in-law, Michael Barry, who previously served as Fresh Market's chief financial officer, is also considering rolling his 6.4 percent stake in the deal, the people added.

No offer to Fresh Market has been made and none may materialize, the people cautioned. Shares of Fresh Market jumped as much as 14 percent on the news.

Fresh Market, Berry and Barry did not respond to requests for comment. Apollo declined to comment.

Based in Greensboro, North Carolina, Fresh Market operates 176 stores in 27 states across the United States. It has a market capitalization of $1.3 billion.

Fresh Market has struggled with falling same-store sales, which it has partly blamed on increased competition. Fresh Market's main rivals in the specialty retail landscape include Sprouts Farmers Market Inc, a former investment of Apollo, Whole Foods Market Inc and Trader Joe's.

Prior to Fresh Market's initial public offering in 2010, the founding family also explored a sale that attracted the interest of several private equity firms.

The supermarket sector has struggled with lower margins and increased competition in recent years, making it more open to consolidation and attracting the interest of buyout firms that see opportunities to cut costs further.

Albertsons Companies Inc, the supermarket operator backed by private equity firm Cerberus Capital Management LP, acquired peer Safeway in January for more than $9 billion. Albertsons delayed its initial public offering indefinitely earlier this week, citing unfavorable market conditions.

(Reporting by Lauren Hirsch and Greg Roumeliotis in New York; Editing by Bernard Orr)

By Lauren Hirsch and Greg Roumeliotis

Stocks treated in this article : The Fresh Market Inc, Apollo Global Management LLC