The 33-year-old Portugal defender has 18 months left to run on his current deal, which he signed in October 2015.
Southampton's director of football Les Reed says the player has had the chance to sign an improved deal, but Fonte has now asked to leave the club.
"He's had several opportunities to improve the contract situation. He's reserved his right not to do that," Reed told BBC Radio Solent.
"He's made it very clear he would like to explore the opportunities for a transfer.
"That's where we are at the moment, Jose wants to leave the club. He's formally asked for a transfer."
Reed went on to confirm the club have not yet received any formal bid for the player, who joined Saints from Crystal Palace in January 2010, and has made 288 appearances.
Fonte was linked with a move to Manchester United in the summer after helping his country win Euro 2016, and he wrote on Instagram last month: "Just to set the record straight I did not reject a new contract. In fact, I have been informed by Southampton that they are not offering me a new contract."
However, Reed insists new terms have been offered to the former Benfica player: "What was offered to Jose was, in my view, quite significant off the back of the contract he signed in October 2015.
"Six months later we were prepared to improve that contract and extend it. He has turned down the opportunity to increase his salary, and he's turned down the opportunity to get another permanent year on his contract."