New York radio talk show host Marc Malusis said earlier this week that the Browns and Vikings were working on a trade that would send Beckham to Minnesota for 2021 second- and fifth-round picks.
"Completely false," DePodesta said.
"It's frustrating a little bit. I think it's pretty clear what we're trying to build at this point.
"We're really building around a corps of players that we think we have a chance to have a championship-calibre core, and the idea we would take away from that core at this moment just doesn't make a whole lot of sense and really not something we are exploring at all."
Cleveland acquired Beckham, a three-time Pro Bowl selection and one of the league's most dynamic receivers, along with pass rusher Olivier Vernon last March in a blockbuster trade with the New York Giants.
The move stamped the Browns as an expected contender in the AFC and a team poised to end a then 16-year playoff absence, the longest active streak in the NFL.
Though Beckham played in all 16 games and eclipsed 1,000 receiving yards for the fifth time in his six NFL seasons, the Browns failed to live up to the hype and finished a disappointing 6-10, leading to the dismissal of coach Freddie Kitchens and rumours of the flashy wide receiver being unhappy in Cleveland.
Beckham later denied those reports about his perceived displeasure, and DePodesta told reporters during Thursday's conference call he's had no indication that the 27-year-old wants to leave.
"I have no reason to believe that he doesn't want to be here," he said.
"Odell's been very good this offseason. He's been engaged I know with (new coach) Kevin (Stefanski). I know he's excited about the possibilities of what this offensive system could bring for him and we're excited to have him."