Evina's current deal at The Valley expires next month and Charlton were keen to tie down the former Arsenal man to a new contract.
However, Evina has rejected the club's offer of a one-year extension and now looks set to leave The Valley.
The 21-year-old admits he was disappointed by Charlton's offer and is looking for a bit more security with a longer deal.
"I have turned down Charlton's offer of a new deal and to be honest I am very disappointed with the club," Evina told Sky Sports.
"At this stage of my career I need more security and unfortunately for whatever reason the club cannot do that for me.
"I now have to consider my future and I am keeping my options open."
News of Evina's possible availability is sure to alert a number of clubs with Reading and Leeds both thought to be admirers of the left-back.