They have been linked with the Brazilian almost ever since they his compatriot and teammate Sandro from Internacional in September 2010.
The striker had scored in the club's landmark Copa Libertadores final win, a competition which convinced them to sign Sandro.
The possibility of bringing him in to White Hart Lane has remained frequently mentioned, and despite him becoming a full Brazilian international and scoring in pre-season against Barcelona last summer, Spurs have so far failed to act.
Now he has scored a superb six goals in four games at the Olympic Games, and with the final still to come, it is little surprise Tottenham are being linked with rekindling their interest.
The problem is they won't be the only ones interested in the 23-year-old.
If clubs and scouts around Europe were not totally convinced by him beforehand, they will be queuing up around the block to sign him now.
Internacional are hardly in desperate need to sell either, having just received £25 million from Chelsea for teenage prodigy Oscar.
Manchester United, PSG, Juventus,and AC Milan have all been linked with his signature, and Tottenham now face intense competition for it when last summer they could have stolen a march to sign the players.
Daniel Levy should have signed the striker last summer, instead of taking the cheap route and bringing in Emmanuel Adebayor on loan at the last minute. They could have sold Modric to finance it if need be, because the Croatian's situation this time around makes it like Groundhog Day.
The Olympics have rocketed Damiao's price tag, and made his signing a lot more difficult than it already was. If Spurs do miss out, they have themselves to blame for not pushing a deal one, even two years earlier, when it seemed easier and cheaper to tie up.
Do Spurs still have a chance of Damiao? Should they have taken the plunge on him earlier?
image: © simonw92