McCoy participated as expected in the team's voluntary offseason conditioning program on Monday just days after Cleveland drafted his likely successor, Oklahoma State QB Brandon Weeden.
The Browns selected Weeden, a former minor league pitcher in the New York Yankees' system, in the first round on Thursday and the team is expected to start the 28-year-old this season as a rookie. McCoy could be kept as his backup and Browns president Mike Holmgren said over the weekend he could imagine McCoy staying on as the team's No. 2 QB.
"That could be a difficult situation except if it is ever going to work, it will work if that happens, because Colt McCoy is a special young man," said Holmgren, who picked McCoy in the third round in 2010. "Of course he wants to play, they all want to play. Again, nothing has been done yet, we don't know how it is going to sort itself out.
"But, if that were to be the case, at some point, I think we have the best chance of making that work because of who the people are, who the players are. It is never easy, everyone wants to play. But, you only have one ball and only one guy can play at a time."
Holmgren insisted Weeden will not be handed the starting job, but the Browns wouldn't have taken him with the No. 22 overall pick -- much sooner than most experts had him going -- if they intended to groom him for the future.
"In this business, you line them up, you give it your best shot and then we have to choose," Holmgren said. "We have to choose somebody to play. Right now, we have four quarterbacks on the roster and they are going to compete for the position."
Seneca Wallace was McCoy's backup last season.
The team has had internal discussions about dealing McCoy, but general manager Tom Heckert said the club has not discussed potential deals with any teams.
It is not known if McCoy has asked for a trade.
McCoy is 6-15 in two seasons as a starter. He made 13 starts last season before missing the final three games after suffering a concussion on a hit by Pittsburgh linebacker James Harrison. The Browns were criticized for their handling of McCoy's injury, and the NFL implemented new policies as a result of the helmet-to-helmet hit.
McCoy's development last season was slowed by the Browns tying for the league lead in dropped passes. However, McCoy doesn't possess the same strong arm as Weeden and coach Pat Shurmur and Holmgren both said they believe better play by the team's quarterback will lead to more wins.
Shurmur said the Browns are better positioned to start a young quarterback, and Weeden's maturity makes it easier.
"I think he has crossed a lot of bridges that will give him an advantage if it happens quickly," Shurmur said. "But, I still think you bring him in, you work with him, you see what happens and then the overriding decision in all of this is that when we play our first game we put our best players out there. I think that's what we'll see."
Meanwhile, the Browns have signed one of Weeden's good friends and top college targets, wide receiver Josh Cooper. He was Oklahoma State's second-leading receiver last season, catching 71 passes for 715 yards and three touchdowns.
Shurmur said the Browns will sign around 15 undrafted free agents.