Taylor, who is out of contract next month, has made 163 appearances for City since arriving from Team Bath in the summer of 2007.
The central defender has played a key role in the club's rise from the Conference to League One but now wants to move on and is in talks with other clubs.
Tisdale thanked Taylor, 29, for the four years of service he has given the Grecians.
"Matt is unlikely to stay with us and we`ve agreed that he will look at the possibilities elsewhere," Tisdale told the club`s official website. "He feels that it is the right time for him and we fully support that, but the door is always open for Matt because he has been a big player for us.
"He has given the club so much and we owe him a lot. He deserves a good move if he can find one."
Source: PA
Source: PA