Mills is not keen, however, to play in the First Division.
His agent Neil Featherby told BBC Sport: "You would have to be naive to think under the present circumstances he is likely to remain there."
Featherby added: "Of course he would prefer not to play in Division One, just as Leeds United would prefer not to play in Division One."
Middlesbrough are probably slight favourites to sign the England international.
Featherby said: "He's been more than happy at Middlesbrough throughout the season, and more to the point he came away with a Carling Cup winners' medal."
Interest from Spurs is likely to grow if right-back Stephen Carr is tempted to join Newcastle.
But the Magpies are rumoured to be interested in Mills if Carr stays put.