Adebayor is a free agent after his short-term deal at Crystal Palace ended, and Flamini is also out of contract after spending a combined total of seven years at Arsenal in two spells at the club.
Caykur Rizespor coach Hikmet Karaman told 'Anadolu Agency': "We reached an agreement in principle with both players.
"They'll come to Rize within a few days and the last step of negotiations are going to be done.
"This is down to everyone devotedly working to finish the deals, so I have to thank the board of directors and our president".
Caykur Rizespor finished 13th in the Turkish Super Lig last season, gaining 37 points from their 34 matches.