Smith suffered a concussion last week against the St. Louis Rams.
"Alex is still going through the protocol right now," offensive coordinator Greg Roman said Saturday. "And that's in the doctor's hands. When we know, you will probably know."
Smith did not speak to reporters Saturday.
For the fourth consecutive day, Smith practiced in a non-contact black jersey and took part in most of the drills involving the first-team offense. The NFL's third-ranked quarterback with a passer rating of 104.1, Smith has received positive reviews throughout the week.
"He's looked good, really sharp," Roman said. "He's gotten plenty of practice time. There's a lot of different scenarios this week and it's fluid. We talk through them, and it will change according to how the big picture changes, and obviously the quarterback position is a huge piece of that."
Smith, who has made 28 consecutive starts, was injured during the first half of a 24-24 tie with the Rams and left the game midway through the second quarter. He was replaced by second-year veteran Colin Kaepernick, who would make his first career start Monday if Smith is unable to play.
Offensive tackle Joe Staley suffered a concussion in Week 6 against the New York Giants, and he went through the NFL-mandated protocol before returning to play the next week against Seattle. Staley said he has seen nothing from Smith in practice this week to suggest the quarterback won't be able to return against the Bears.
"He was normal as far as going through all his tests," Staley said. "He just has to go through the final steps."