The Brazilian striker has supposedly been on the verge of a move to the Blues for two years or more, yet he remains on the books of Portuguese champions FC Porto.
His full name may be Givanildo Vieira de Souza, but his nickname is fully warranted. A muscle-bound brute of a striker, Hulk is often pegged as one of the candidates to be Didier Drogba's true replacement in west London.
But do the European champions really need Hulk on their books in order to once again become a genuine title-winning contender? Should they pay the huge asking price to take him from the Estadio do Dragao to Stamford Bridge?
A lot has been made of the £78 million buyout clause in Hulk's contract, but in truth, he would cost around half that were he to be sold this summer.
However, that is still a hefty price to pay for a club which has already spent around £40 million on Eden Hazard and Marko Marin, and they are also trying to sign Andre Schuerrle. With UEFA's financial fair play regulations to comply with, can Chelsea afford to gamble so much money on a single player who has never produced against English opposition?
Hulk's record is hugely impressive. In the 2010-11 season, which ended with Porto winning the treble, he scored 36 goals in 53 games. Last season, he missed around half the campaign through injury, but he still managed to score 20 times in 33 appearances.
However, he has never scored in 10 matches against opponents from the Premier League, having started nine of those encounters.
Should Chelsea buy Hulk?
Former Chelsea assistant manager Ray Wilkins was in the dugout when the Blues played against Porto, and he was not particularly impressed by their Brazilian striker.
"I was at Chelsea on a couple of occasions when we played against Hulk," Wilkins told the Fulham Chronicle. "He’s not in the class of Drogba, but then few people are."
The player himself seems rather ambivalent about the prospect about making the move to London, having been too coy in barely even acknowledging a deal that the world and his wife has known was on the table for a long time.
He is quoted by Metro as saying: "I have four more years on my contract with Porto, and I'm focusing on the national team.
"I haven't talked to anyone. As everyone knows, I don't follow negotiations closely as that will make me lose my focus. I let my agent deal with that stuff."
The affair has dragged on for so long that even Chelsea are losing patience, with the Evening Standard reporting that a £16.7 million move for another Brazilian forward, Shakhtar Donetsk's Willian, is back on the agenda.
With Drogba gone, Fernando Torres is very much the main man in the Chelsea attack now. The Spain striker has still yet to fully sparkle during his 18 months at Chelsea, but he will return from a summer in which he won Euro 2012 as the tournament's Golden Boot to find at least two new creative attacking teammates employed to lay on the chances for the club's record £50 million signing.
The presence of Daniel Sturridge on the right of a three-man unit behind centre-forward Sturridge would add a degree of versatility to the Chelsea attack, and the England striker could even take his preferred position up front himself if Torres is rested, injured or, heaven forbid, out of form.
With the club yet to decide whether or not to send Romelu Lukaku out on loan next season, the young Belgian striker could be another attacking option if he is deemed worthy of a place in Roberto Di Matteo's squad.
Hulk has been one of the most talked-about names in each of the last few transfer windows, but in terms of making a move to Chelsea, that ship may have sailed.