The 27-year-old missed his side's 4-0 win over St Johnstone this afternoon having suffered a recurrence of a calf injury which he has been unable to shake off so far this season during the midweek Champions League defeat to Bayern Munich.
Griffiths will now play no part in the first match under interim manager Malky Mackay, leaving Nottingham Forest's Jason Cummings as the only out-and-out striker in the squad.
"He won't play. He had a slight calf injury from our game during the week. He had a scan and it's not overly serious, but it will probably be a week to 10 days, which is unfortunate," Rodgers told reporters.
The SFA has also confirmed that Darren Fletcher will miss the match despite playing the full 90 minutes of Stoke City's 2-2 draw with Leicester City this afternoon.