Godwin also could miss the Buccaneers' game against the Chicago Bears in Week 5, which is scheduled for a Thursday night and would be a quick turnaround from Week 4, a source said.
The Buccaneers hope Godwin can return no later than their Week 6 game against the Green Bay Packers, the source said.
"It's the 'next man up' at that position," Buccaneers coach Bruce Arians said. "We've got guys. Tyler [Johnson] got his first action last week, he's had a couple good weeks of practice. We've got Justin [Watson] back, who's had a heck of a camp. It's not like we're empty. So we've got enough guys -- [Jaydon] Mickens, [Cyril] Grayson -- we've got enough guys to fill the void at wide receiver. I'm not as concerned with that as I would be if we lost one more."
Wide receiver Scotty Miller did not practice Wednesday with a hip and groin injury.
A source said the Buccaneers are also monitoring the status of starting cornerback Sean Murphy-Bunting, who suffered a hamstring/groin injury against Denver and did not return. Arians said Monday that Godwin and Murphy-Bunting were both undergoing MRIs.
Running back Leonard Fournette also did not practice Wednesday with an undisclosed injury.
Arians said Monday that he believes the lack of a preseason is partially responsible for a number of soft-tissue injuries.
"I think that had a big part of it. The whole league [has experienced] soft-tissue injuries and major injuries," Arians said. "The lack of spring practice and the lack of training camp, I think you're starting to see that now. Just continue to work and do as much. Our sports science people are excellent and try to make sure they don't happen again."
It has been an injury-plagued start to the season for Godwin, who emerged as one of the NFL's top receivers in 2019 when he hauled in 86 catches for 1,333 yards and nine touchdowns. He also missed Tampa Bay's Week 2 game against the Carolina Panthers because of a concussion and has 11 receptions for 143 yards and a touchdown this season.
Godwin was among a number of Buccaneers players to suffer hamstring injuries last season, prompting the team's sports science staff to reexamine how those injuries were handled.
ESPN's Jenna Laine and Dan Graziano contributed to this report.