Morgan underwent surgery in Gloucester on Sunday night to re-set an open dislocation on the small finger off his right hand. The finger was not fractured as first feared and he suffered no tendon damage around the knuckle.
"At this stage it is too early to say when he might play again and we would not make a decision about this weekend's game against Cardiff for another couple of days to let the injury settle down," said head coach Dean Ryan.
"But we are obviously extremely relieved that the injury was nowhere near as serious as first indicated because Olly is a key member of our squad and has been in outstanding form.
"He can begin his rehab almost immediately and with the number of games we have approaching it is vital we have players of Olly's calibre around the squad."
Gloucester's medical team will also monitor Mike Tindall's dead leg and he will continue to receive treatment this week ahead of their Heineken Cup game against Cardiff on Sunday.