Roark has been battling it out with a group of rival private-equity firms including TDR and Sycamore, and pulled ahead in recent days, the report said, adding a deal could be finalized this week.

Subway in February said it was exploring a possible sale of its business. Private equity firms Roark Capital and Advent International, and Goldman Sachs' asset management arm, were among bidders for the company, media reports later said.

Subway and Roark Capital did not immediately respond to Reuters requests for comment.