A source told ESPN that Bryant -- three days after suffering the injury -- had an MRI on Wednesday morning, which revealed a tibial plateau fracture to his right knee.
Another source told ESPN NFL Insider Adam Schefter that the injury could sideline Bryant up to three weeks, depending on how the receiver feels. He did not practice Wednesday, but Garrett said he worked out and "we're hopeful that he'll be able to play."
"What I've been told that it is a day-to-day, week-to-week injury," Garrett said. "We've had players with this kind of injury who missed no time in the past, and other players who have missed up to a week, two [weeks] or more than that.
"We'll take it day by day, and we'll see how he does when he comes in [Thursday], see how he feels, and we'll make our evaluations there as we go forward."
Garrett would not say whether the players who have had similar injuries in the past were skill position players, but a source said Bryant's position could mean a longer recovery.
Bryant missed only six plays after he was injured on the second play Sunday against the Chicago Bearswhen linebacker Christian Jones dragged him down after an 11-yard gain. Bryant's knee and ankle got caught under Jones, and he went to the ground before getting off the field.
He finished the game with three catches for 40 yards, including a 17-yard touchdown in the fourth quarter. Bryant played 52 of 68 snaps but was not on the field late in the fourth quarter as the Cowboys attempted to run out the clock.
That he did not have the MRI until Wednesday morning led to confusion regarding Bryant's status, but Garrett would not clarify why it was delayed.
"I don't want to get into the whole medical procedures," Garrett said. "We have this information now with Dez, and we're going to take his situation day by day. We'll evaluate his status for the game as the week goes on."
The Cowboys' offense was stagnant last season without Bryant and quarterback Tony Romo, who missed 12 games with a twice-broken left collarbone. Romo is currently recovering from a compression fracture in his back suffered in the preseason and is improving but not ready for practice.
If Bryant can't play Sunday, then Brice Butler, who has one catch for 16 yards in three games, would move into the starting lineup. Rookie Dak Prescott said Bryant's potential absence does not affect the Cowboys' plans.
"I don't really pay attention to who I'm throwing to," said Prescott, who has connected with Bryant 11 times for 150 yards and a touchdown this season. "I just go through my reads, whoever's open gets the ball. So I mean hope [he] gets well, is well really, really soon, but it doesn't really affect this offense."
ESPN Staff Writer Nick Wagoner contributed to this report.