The Mirror says they want to sign Trafford in January as cover for injured No1 Guglielmo Vicario.
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For the moment, Fraser Forster has been acting as the stand-in, with Spurs keen to bring in a younger backup for the long-term.
Trafford is also interesting Newcastle for the January market.
Like with Spurs, the England U21 keeper would arrive as cover for No1 Nick Pope.