The Russian giants travel to Glasgow for the first leg of a last-32 Europa League tie with Celtic and Rodgers's side are big underdogs to reach the next phase of the competition.
Worryingly, the Scottish Premiership leaders haven't been at their best domestically for quite some time, grinding out a nervy 3-2 win over Partick Thistle last weekend after losing to Kilmarnock 10 days ago.
Zenit have the personnel to pounce upon any mistake made by the Hoops tomorrow night and, according to Rodgers's assistant Davies, the Northern Irish manager has been stressing the importance of remaining switched on for the entire 90 minutes at Parkhead.
He told The Daily Mail: "The manager has been reinforcing to them the concentration aspect. It's really important at this level. When you are facing teams like Zenit it can be about that mental fatigue," he told The Mail.
"You need to concentrate, be switched on and alive, for every single moment of the game. That's going to be very important on Thursday to ensure we are defensively solid."
It's a smart message by Rodgers, whose side have been prone to the occasional lapse of concentration on an increasingly-frequent basis.
Celtic fans won't mind elimination providing that Zenit are far and away the better side over two legs, but what will frustrate the Parkhead faithful is if Rodgers's side shoot themselves in the foot and ship soft away goals to make the return fixture a mere formality for Roberto Mancini's troops.