Spurs are long-term admirers of the 23-year-old, with Internacional having previously claimed they turned down a bid from the Premier League club in the January transfer window.
The highly-rated Brazil international is also thought to be attracting interest from Italy, with Napoli reportedly targeting him as a potential replacement for Edison Cavani, should he leave at the end of the season.
Damiao is currently contracted at Internacional until 2017 and reports in Brazil suggest the club would only part with him for upwards of €20million.
The Brazilian side's director of football, Luis Cesar Souto de Moura, admits it is expected he will be prised away in the summer due to the amount of interest being shown from abroad.
He told UOL: "Leandro Damiao is indeed the player that is most likely to leave Internacional, especially because foreign clubs are interested.
"Damiao is the player of the moment, everyone is seeing that. He is a great player, he's very special."