The sale of Cygnet, which has 17 UK-wide mental health and addiction centres, could cost approximately £200 million.
Cygnet had increased revenues in 2013 of £90.9 million, up from £81.4 million the previous year. Its earnings before interest, taxes and depreciation (ebita)stood at £23.1 million for the same financial year.
The Priory recently failed to buy complex care provider Voyage Care, losing out to investment firms Duke Street, Partners Group and Tikehau who collectively paid £375 million.
Since Advent International bought the Priory in 2011, it merged the company with mental healthcare provider Craegmoor and also backed the purchase of Peninsula Autism Services & Support in May 2012.
Cygnet, the Priory and Advent have so far declined to comment about the potential deal.