Winn injured his knee and ankle during Saturday's practice.
The Browns said the ankle is the more severe of the two issues, and Winn will be evaluated on a weekly basis.
In addition, the team had so many running backs sidelined it may have to add another back.
Coach Mike Pettine said Johnson's injury may linger for a few days. That's not good, as Johnson was expected to be an immediate contributor.
Pettine said the hope is West will return after Monday's day off.
Not participating: Johnson, West, Winston, CB Ifo Ekpre-Olomu (knee), CB Brandon Stephens (hamstring), TE Randall Telfer (foot) and Winn. Lundy and OL Vinston Painter (neck) left practice with injuries.
Camp tidbit: Receivers coach Joker Phillips met the media on Sunday, and Phillips emphasized that there are several elements of the receiver position he does not need to teach Terrelle Pryor. Those include assignments, adjustments and alignments, because Pryor knows John DeFilippo's offense. Phillips also said Pryor has excellent hands, so he doesn't need to teach ball skills. "All I can coach is technique on the young man, and he's really willing," Phillipps said. "When I talked to him right after we signed him, he was already working with some quality guys. Now, I'm just trying to teach him our techniques and how to play the game."
From the coach, on Pryor: "He doesn't look out of place. … He does some good things. He is just behind from the learning standpoint and the subtleties of playing receiver. If you had asked where he would be at this point, I would think we would all agree he is ahead of where we thought he would be." — Pettine.
Monday: The Browns have their first day off in training camp, so there is no practice or player interviews. The Browns return for three days of work in Berea, Ohio, before heading to Columbus for an intrasquad scrimmage Friday night.