The sandwich company sells to retailers ranging from Boots to Costa Coffee, and started a search in August to bring in a new investor to expand into new territories in the UK.
According to The Telegraph, Towerbrook, a New York-based firm, along with a range of other private equity firms, are pursuing the company to buy half of the business.
The newspaper said it understood that second-round bids had already been lodged by the private equity firms.
Adelie currently operates six production facilities across the UK that combine to produce 4m fresh sandwiches and 250,000 salad packs each week for distribution.
It is the second biggest sandwich firm in the UK, after Greencore which hit the headlines last week for turning to Hungary for workers instead ok finding Britons.
It owns brands Urban Eat and Cranks, and distributes to food-to-go customers.