Both Antonio and team-mate Andy Carroll were left out of the squads for pre-season trips to Austria and Germany, but are fighting for full fitness.
"Michail and Andy are in London," said Bilic.
"He [Antonio] is ahead of the plan."
"The plan is for him to miss the first game of the season."
However, on Carroll Bilic was less optimistic, adding.
"Andy is still not able to train with the team but he is working hard at Rush Green with the physios."
"I am frustrated he cannot play, now he is still injured and it is a big loss for us."