Spurs were interested in the summer, but the Toffees turned them down, as did a £40 million proposal from Chelsea.
Gordon, 21, has three goals this season and is a big supporter of Everton manager Frank Lampard.
Lamps is still desperate to keep the England Under-21 international at Goodison Park in the long run.
However, there are fears around the club that the summer attention was a distraction.
Spurs manager Antonio Conte is considering testing Everton’s resolve in January, and preliminary inquiries have already been made.
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Conte was looking for another creative player in the summer, and Spurs were struggling after Dejan Kulusevski’s injury.
Lampard’s attachment to Gordon, a homegrown talent, implies that a deal will come at a high cost.
Everton paid Tottenham £60 million for Richarlison, and Goodison officials will demand a similar price again.
Everton have begun contract talks with Gordon in an attempt to ensure his future, but no agreement has yet been reached.
Chelsea, under new manager Graham Potter, are not expected to rekindle their interest in the January transfer window.
This might give Spurs a free run at him.
Despite concerns about his concentration, Gordon has always maintained that he is not in a hurry to depart.
