The frontman was loaned to the Hammers for the season at a cost of 5million euros, while they hold the option to make his switch permanent for a further 20million euros.
Zaza has struggled for form since arriving in England leading to suggestions that West Ham may opt to send him back to Italy.
However, Bilic insists there is no truth in the current rumours and expects him to eventually come good once he finds his feet in new surroundings.
Bilic said: “This one is false. He’s our player, we got him, he’s here. He’s training good and we expect him to do the job.
“I have to make a decision on Zaza over Saturday. That is the only decision I am thinking about: the game against Palace.
“The only thing I am thinking about Saturday is not about the situation in his contract or around his contract but is he going to be the one I will put at centre-forward? The only objective is: can he do it against Palace?”
Southampton have announced plans for a two-week tour of China as part of their pre-season preparations for the 2018-19 season.