The Brazilian is currently struggling to nail down a spot in the first-team and he is reportedly growing frustrated.
A number of clubs have been alerted to the 27-year-old's situation and his agent has received several proposals.
Everton, according to Calciomercato, are leading the race for Neto and could snap him up during the January transfer window.
The Toffees signed Maarten Stekelenburg from Fulham in the summer, but Ronald Koeman is said to still be on the hunt for another goalkeeper.
Juventus, however, are trying to persuade Neto to stay put as they view him as the long-term replacement for 38-year-old Gianluigi Buffon.