Griffiths has been sidelined for around four weeks with a calf strain sustained before half time in the win over Hibernian on Saturday afternoon.
There is even worse news for goalkeeper Gordon, meanwhile, as he suffered a knee injury in the same game and is unlikely to feature again for three months.
Rodgers confirmed the double injury blow in his pre-match press conference ahead of Tuesday's home match against Hearts, and outlined his intention to bring in goalkeeping cover before the January transfer window closes.
"It is sad news for [Gordon] and for us but of course we will need to have that cover for someone to come in and support Dorus de Vries so that is something that we are looking at," he told reporters.
Celtic face Zenit at Parkhead on February 15 and then travel to Russia a week later for the second leg of the last-32 showdown.