Bilic led Beşiktaş to a memorable win over Liverpool in the Europa League Round of 32 to eliminate Rodgers side from the competition.
The Black Eagles lost 1-0 against Liverpool at Anfield but beat the Reds on penalties at the Atatürk Olympic stadium – the same venue Liverpool won the 2005 Champions League.
Beşiktaş were eventually knocked out by Belgium outfit Club Brugge in the quarter-finals.
Bilic guided the Istanbul giants to third place in the Turkish league last season before resigning from his position and replacing Sam Allardyce as West Ham boss.
The Hammers are currently event in the Premier League on six points after four games and take on Newcastle United next at home.
Liverpool dropped into sixth place on seven points following the defeat and take on Manchester United at Old Trafford next.