With Emmanuel Adebayor still on international duty with Togo at the Africa Cup of Nations, Jermain Defoe is currently Tottenham's only fit senior striker.
That has prompted speculation Spurs boss Andre Villas-Boas could bolster his forward line in January, with 23-year-old striker Damiao believed to be top of his shortlist.
Damiao, rated at £20million, has impressed since joining Internacional in 2009 and was Brazil's top scorer at the London Olympics.
He has previously indicated that he was happy to stay at the club until after the World Cup in 2014, though would not completely rule out a move to Europe.
Internacional's president Giovanni Luigi has insisted no bid has been lodged by the Premier League club for Damiao, who has two-and-a-half years left on his contract.
However, he concedes there remains a possibility he could leave before Thursday's deadline, though talks would have to be swift.
"Leandro Damiao will only leave if there is a very quick negotiation," he told Correio do Povo. "But no official offer has been made."
Internacional director of football Luis Cesar Souto de Moura confirmed Tottenham's interest in the striker, but thinks it is unlikely any deal will be completed in January.
"The news involving interest for Leandro Damiao, especially from Tottenham, will always happen, they come and go," he told Terra.
"Damiao has a lot of quality, is a goalscorer, was the top scorer in the Olympics and he is a very important player.
"Internacional understand that it's extremely difficult, very difficult, for Leandro Damiao to leave now."