Capricorn, which bought The Real Greek owner Clapham House for £30.4m last September, is understood to have received interest from three parties for the chain, which is led by managing director Liz Williams.
A trade buyer is believed to be among the suitors, although a management buyout has not been ruled out.
London-based The Real Greek operates sites in Bankside, Covent Garden, Hoxton, Marylebone, Spitalfields and Westfield. A seventh site in Putney was disposed of last year.
Final bids for the chain were submitted last week and a due diligence process is under way, with a deal expected to be completed in May.
A sale of The Real Greek had been mooted by Clapham House as far back as 2009.