The Grecians announced the quintet of midfielder Tommy Doherty, defenders Steve Tully and Myles Anderson and strikers John O'Flynn and Tom Nichols have been handed fresh terms to stay on at St James Park.
However, 35-year-old midfielder Oakley and 37-year-old striker Cureton, who made over 70 appearances between them last term, could walk away from Exeter despite ongoing talks with the club.
Chief executive Julian Tagg told the club's official website: "We continue to live within our means and will be acting in the interests of the long-term health of our club as a whole this summer, which places extra emphasis on value.
"But we retain optimism in our outlook as we continue to work hard to tie up contracts."
Last month boss Paul Tisdale confirmed goalkeeper Rhys Evans, defender Kevin Amankwaah, midfielder Mark Molesley and striker Guillem Bauza would not be retained and winger Elliott Frear has now been informed he will not be offered a new deal either.
- Sports & Recreation
- Myles Anderson
- Tommy Doherty