They are still in pole position to sign him, PSG have thrown their hat into the ring, but with the deal still in limbo, this would be the perfect time for Spurs to make a move.
It would be the ultimate test of owner Daniel Levy's ambition.
The man who sacked Harry Redknapp has been criticised for being tight with funds over the past two seasons; signing a superstar like Hulk would change perceptions dramatically.
Anybody asking whether Spurs have the money to do so, should consider that they are likely to receive around £30 million from the sale of Luka Modric.
Yes the Croatian will have to be replaced, but with Gylfi Sigurdsson on his way, they are already half there. Links with Joao Moutinho have surfaced through the Villas-Boas connection, and it is unlikely Spurs would sign both him and Hulk given the expense required.
Spurs would do well to make Hulk a priority. In attack presently they have just Jermain Defoe; Gio Dos Santos is reportedly off, and Adebayor's return currently looks doubtful.
25-year-old Hulk flourish under the management of Andre Villas-Boas at Porto. The manager was in charge for just one season, leading the club to the Europa League and the Portuguese domestic league and cup.
In that one season in 2010-11, Hulk scored 36 goals, and provided 21 assists - an astounding record yielding 57 goals in just one season.
If AVB can get him playing anything like that - last season it was 21 goals and 15 assists - he can fire Spurs back into the Champions League places.
Does Champions League football matter to Hulk? Ideally he would like to play in the competition of course, but the last two years it has been Europa League football with Porto. The chance of linking up again with Villas-Boas may compensate.
At Tottenham he would become an automatic first choice, over at Chelsea the situation is a little more unclear with Fernando Torres an enigma.
The longer goes with Chelsea not tying up a deal, the more chance there is of another club putting in a £30 million bid plus to secure his services.
If Tottenham are brave and ambitious enough, it could just be them.
Should Tottenham go for Hulk? Would they?
image: © *dirk