Bremer signed a new contract at Torino earlier this year but it was largely done to protect his value for the Italian club.
The Brazilian defender has been expected to move on from Torino in the ongoing transfer window with several clubs interested in him; Inter are currently the favourites to snap him up.
Tottenham have their eyes on him as Antonio Conte pushes to bring in a centre-back and Vagnati admits that Torino are entertaining several approaches.
The Torino sporting director insisted that a decision will be taken after consulting with Bremer.
Vagnati told Sky Italia: “Bremer is a great defender and an exceptional guy – a leader and an exemplary professional.
“We have interest from many and we must try to understand with the boy, which is the best path.”
Inter have been tipped to be the favourites to sign Bremer and are said to have agreed on personal terms with the defender.
However, Vagnati stressed that claims of Inter being close to signing the Brazilian are exaggerations at the moment.
“If we say that he is close to joining Inter, is it an exaggeration? Yes.”
Bremer is a defender Tottenham could turn to after being snubbed by Inter’s Alessandro Bastoni.