The midfielder - who has scored nine times this season - came off just 15 minutes into the 1-1 draw at Barnsley on Saturday after sustaining ankle and knee damage in a tackle.
Norris has been sent for a scan to determine the severity of the injuries but he is unlikely to recover in time to make another trip to Yorkshire.
Colin Healy replaced Norris at the Keepmoat Stadium and could start in his place if required.
Striker Jason Scotland could also come into the starting XI. Scotland has been talking up a move back to former club Swansea but came off the bench to score at Oakwell.
Goalkeeper Brian Murphy has stepped up his comeback from a broken ankle after coming out of plaster and will be back in around six weeks, meaning Marton Fulop will continue.
Source: PA
Source: PA