A deal may be announced any time, Bloomberg said, citing three people with knowledge of the matter, who spoke on condition of anonymity because the talks are private.
Florida- based Acosta is one of the biggest U.S. marketers to food and grocery stores.
Three sources familiar with the matter told Reuters, a sister news service to Buyouts, that Acosta has been up for sale.
Acosta has been running a quiet sale process limited to a small group of buyers instead of a broad auction, one of the three sources said.
Acosta and AEA did not return calls seeking comment after regular business hours. THL was not immediately available for comment.
Megan Davies is a Reuters correspondent in New York. Soyoung Kim and Dhanya Skariachan also contributed to this report.