The firm joins rival bidders including Goldman Sachs' asset management arm, Bain Capital, TDR Capital - the joint owner of Asda - and TPG, according to the report.
Advent had previously invested in entertainment retailer HMV and currently has investments in companies such as Laird International and McAfee Corp, among others.
'Exploring A Possible Sale'
Subway noted in February that it was exploring a possible sale of its business amid surging costs and rising competition from deeper-pocketed rivals.
Advent International declined to comment, while Subway did not respond to a Reuters request for comment.
The Wall Street Journal in January reported that a potential sale could value Subway at more than $10 billion.
The Subway Story
The story of the Subway brand started more than 50 years ago when Dr. Peter Buck, a nuclear physicist, changed the life of a college student with a few simple words, “Let’s open a submarine sandwich shop.”
According to the company's website, in 1974, the duo owned and operated 16 submarine sandwich shops throughout Connecticut.
Realising they would not reach their 32 store goal in time, they began franchising, launching the Subway brand into a period of 'remarkable' growth which continues to this day.