Wright, one of a number of ex-Truro City players with Taunton, is considering an offer to play in Sweden.
Midfielder Matt Buse is also looking at his options.
Taunton boss Rob Dray: "We're still in talks with them. "We're in no rush and we don't want to put pressure on them.
"We want them both to stay and I'm confident they will remain with us."
Taunton have signed a number of their players on extended contracts.
These include goalkeeper Lloyd Irish, Pierce Mitchell, Dan Sullivan, Ben Adelsbury, Ryan Brett, Ollie Chamberlain, Ed Palmer, Rocky Neal and Shane White.
Josh Nelmes has also signed a contract which will keep him at the Viridor Stadium until the end of the season.
Dray added: "It's important to have these boys signed, as they are crucial to our medium to long-term ambitions.
"They're proved what they're capable of, they're great characters and it means they're settled going into the summer".