Tampa Bay dealt Winslow to Seattle for a draft pick on Monday night and signed former Colts tight end Dallas Clark.
Winslow did not participate in a voluntary workout with the Buccaneers. He said in a radio interview Monday that Schiano had told him the team would likely trade the 28-year-old former first-round pick.
And a few hours later, Winslow was headed West. Tampa Bay received a conditional seventh-round draft choice in next year's draft for Winslow.
"There is really not a message. It's not one of those situations," Schiano said Tuesday. "The only message I want to address is we need to be the best football team we can be. Time is our enemy. We are running out of time. Teams in our division have a head start on us."
Winslow was the NFL's highest-paid tight end when the Bucs signed him in 2009. He never missed a game in three seasons and last year caught 75 passes, seventh most among NFL tight ends.
"Kellen (Winslow) is no longer a Buc," Schiano said. "I'd like to focus on the guys that are here. When we do make decisions, I am going to hesitate to look back and really just talk about guys that are Bucs right now."
The 32-year-old Clark has been one of the NFL's most productive tight ends for nine seasons with Indianapolis.
But since his spectacular 100-catch, 10-touchdown season in 2009, he has been limited by injuries. He played 11 games last season and made 34 catches for 352 yards.
"Injuries are part of the game and I've had a couple of down years, if you will," Clark said. "But it's not slowing me down and I'm not feeling like I'm hampered by anything. I still feel I can perform every day and that's what I'm working hard to do."
Since he was on the free agent market for so long, Clark was bothered by the perception that he was finished as a player. He had two other offers, he said, before signing his one-year contract with Tampa Bay.
"No one asked me if I was done playing and I'm not done playing, so I don't know where that came from," he said. "Just because I'm not part of a team doesn't mean, I'm done playing. As for the proving part, you're an NFL football player so every day you're having to prove yourself. The second you stop proving yourself, you're out these doors. You learn that day one."
Schiano said the Bucs did their homework before replacing Winslow with Clark.
"I think he is definitely a proven player," Schiano said. "He has had some health issues as of late, some things that have kept him from playing at the level of football he played earlier in his career, but I believe wholeheartedly that he is going to return to his form and play the way he is capable. I think he can do everything we are going to ask him to do, and it is really good to have strong veteran presence in that tight-end room as well."
Luke Stocker, who was drafted by the Bucs in the fourth round in 2011, is expected to bid for serious playing time at tight end this coming season. The Bucs also drafted tight end Drake Dunsmore out of Northwestern in the seventh round.