"He's day-to-day right now," coach Bruce Arians said Wednesday, according to ESPN's Jenna Laine. "He's got a little bit of a brace on it right now, and we'll see how it goes during the week."
Winston briefly left Tampa Bay's Week 14 win over the Indianapolis Colts in the third quarter but was able to return, throwing another two touchdowns and finishing with a career-high 456 passing yards. He suffered the injury when his hand hit the helmet of a Colts defender just before halftime.
The 25-year-old - who's in a contract year - saw a specialist Monday and was given the all-clear to continue playing.
"It's getting better every day," Winston said. "I'm just taking it day by day. I'm gonna be ready to go as soon as I can."
Winston leads the NFL in interceptions this season with 23, but he's also second in both passing yards (4,115) and touchdowns (26).