Coach Jim Harbaugh said Monday that Morgan was scheduled for a "procedure." The receiver was injured late in the fourth quarter of San Francisco's 48-3 victory over the Tampa Bay Buccaneers on Sunday.
"It's more serious than what I thought after the ballgame yesterday," Harbaugh said. "Josh is going to be out for an extended time, longer than we anticipated and longer than we hoped -- longer than you can afford to have him out. ... I just feel sick for him and for our team. There's nobody you can afford to lose. Especially him, Josh was really playing well."
Morgan caught a 19-yard pass from backup quarterback Colin Kaepernick on fourth-and-3 and was tackled just short of the goal line -- one of his five receptions Sunday.
Morgan grimaced in pain and stayed on the ground for several minutes as teammates gathered around in support. Fans at Candlestick Park chanted "Morgan! Morgan!" before he was carried off by tight end Vernon Davis and trainers and then carted to the locker room.
Harbaugh isn't counting out Morgan returning late this season or for a potential playoff game.
"That's a hope. We'll have a better idea after today, tomorrow and see where we're at to assess the situation," the coach said. "Josh was really coming into his own, catching his stride. I feel bad for him, feel bad for us. That's a tough loss for us."
Morgan, a fourth-year pro out of Virginia Tech, has 15 catches for 220 yards and one touchdown for the 49ers (4-1).
"You've got to play," Harbaugh said. "You can't take a knee with four minutes left in the game. ... Kaepernick is competing; he's got to get work. We're not taking a knee with four minutes left in the game."
The 49ers are thin at wide receiver, with Braylon Edwards (knee surgery) already out until at least Oct. 30. Michael Crabtree, Ted Ginn and Kyle Williams are the only other receivers on San Francisco's active roster.
The Associated Press contributed to this report.