The 25-year-old has yet to feature for Sarries this season after picking up the injury over the summer, and while he has returned to training, he will undergo a further scan on Thursday.
England head coach Eddie Jones is still expected to name Farrell in his squad for the November internationals on Friday, although the extent of his participation in training remains unclear.
"It's really a day-by-day thing with Owen at the moment - it's a bit of an open-ended injury which is always frustrating for the player, for the media, for me," Saracens coach Mark McCall told BBC Radio 5 live.
"[England] are clearly aware of the situation. I don't suspect he will do an awful lot [in the England camp].
"He's been able to work on his skill work, the one thing he hasn't been able to do a lot of is his kicking."
England face South Africa when their autumn Test series begins on November 12.