Saint Etienne midfielder Bodmer and Brest defender Mendy have been in the position since 2009, when they formed the club by merging two local sides in the Normandy town.
But following disagreements with the board, both have relinquished their posts citing irreconcilable differences with the club.
“Our decision to quit stems from a lack of internal understanding, a permanent state of disagreement, a lack of investment, lack of passion within the club and poor internal communication,” the players said in a statement.
“Initiatives were taken without consulting management and, while we love the club, we cannot continue.”
Former Lyon utility defender Bodmer is on loan at St Etienne from Paris Saint-Germain, while ex-PSG full-back Mendy joined Brest this season after spells at Bolton Wanderers and Hull City.