The Brazilian striker joined Zenit in a mega-money move from Porto in the summer but has struggled to find his best form and did not shake hands with Zenit boss Spalletti as he was substituted during Tuesday's Champions League match against AC Milan.
He later revealed that he may have to consider his future in January, with a number of clubs in Italy reported to be interested.
"I feel well in St Petersburg, my family is happy and my son, too," he told Mediaset Premium Calcio.
"But if the situation with Luciano Spalletti doesn't improve, I might leave in January. I don't know if Italy will be my destination, but anything can happen."
Spalletti responded to Hulk's comments by insisting he has the right to make changes if a player is not performing.
"He can say what he wants," said Spalletti. "He can be disappointed, but if I substitute him it is because I am not satisfied with his performance.
"Has he said he wants to go? I don't know what to do about that. I'm staying here."
Spalletti has a contract with Zenit until 2015 and is paying no attention to rumours linking him with AC Milan, although he admits he would love to manage in the Premier League one day.
"I still have a great desire to work in Russia. I have extended my contract with Zenit and I'm in tune with the president," he explained.
"Me to AC Milan? Now I'm too old for it. But I would like another experience abroad, maybe in the Premier League."