Spurs have tipped off the striker's camp that they would be interested in a loan move if the Frenchman is not sold.
Gameiro was a £25million target for Swansea but he told the Welsh club he was not interested despite the best efforts of top deal-fixer Jorge Mendes.
Turkish side Fenerbahce also want to buy the 30-year-old but he wants to wait to see all his options — and that now includes Spurs.
Atletico would rather sell Gameiro as they have too many forwards and he is at an agreement where his transfer value will start to drop dramatically.
The player would have to be willing to cover and compete with red-hot Kane.