Jay Glazer of Fox Sports reported Sunday the three-time Pro Bowler has told players and coaches to "come get me" before and during games.
One source told Yahoo Sports' Charles Robinson that Beckham made the comments during "at least" one October game.
The Browns faced three opponents in October before the trade deadline: the San Francisco 49ers, New England Patriots and Seattle Seahawks. All three are contending teams that spent the year seeking receiver help.
Each organization upgraded the position in other ways, with Emmanuel Sanders and Mohamed Sanu moving in trades and Josh Gordon added through waivers. However, Beckham could've been an even bigger game-changer for each of the squads hoping to win a Super Bowl title.
As Robinson noted, the 49ers are an especially interesting destination thanks to his relationship with quarterback Jimmy Garoppolo.
"He talks to Jimmy on and off the field," one source said. "They have a friendship that is more than just saying hello before or after [a game]. There's a lot of mutual respect there as football players and people—which is important [to Beckham]."
He also holds 49ers head coach Kyle Shanahan in "high esteem."
Sanders has helped San Francisco since joining the team, totaling 407 receiving yards and three touchdowns in seven games, but he will be a free agent this offseason. The organization could replace him with Beckham next year and give the team a true No. 1 option in the passing attack.
With that said, Beckham is under contract for another four years, and the Browns front office will decide his future.