According to AS, Gaitain is on the north London club's radar this month as they bid to reinforce their squad for a prolonged title tilt.
Mauricio Pochettino is said to be a big fan of his Argentine compatriot, who has a long history of being linked to Manchester United.
The 28-year-old winger only joined Madrid in the summer, ending a five-year stint at Benfica, but he is yet to fully establish himself in Spain.
Gaitan has started just three La Liga games so far this season which has led to suggestions that Spurs may look to offer him an escape route.