Adam Schefter reports the latest on what the Giants are seeking in return for receiver Odell Beckham Jr. (0:51)

The New York Giants have asked for at least two first-round picks in return for Odell Beckham Jr., a league source tells ESPN's Adam Schefter, even though the team insists it is not shopping the three-time Pro Bowl wide receiver.

General manager Dave Gettleman on Tuesday addressed the speculation about Beckham, refusing to directly say that Beckham wasn't available in a trade but also repeating "You don't quit on talent," a phrase his mentor Ernie Accorsi taught him.

There were two recent reports: One, by NFL Network, said that Beckham would not take the field this year without an extension to his current contract. Another, by the New York Daily News, said the Los Angeles Rams were pursuing a trade for Beckham.

"Neither Odell nor his agent have contacted us regarding either report," Gettleman said Tuesday at the NFL owners meetings. "So to be clear: I'm not going to respond to questions about either report, and as I stated earlier, every decision I make is going to be in the best interest of the New York football Giants."

Co-owner John Mara said Sunday that no one on the team is untouchable and declined to say whether he thought Beckham would be on the 2018 roster.

Later Tuesday, Mara said, "He's not on the block. Is that going to stop clubs from possibly calling us? No. But he's not on the block. We're not shopping him around.

"I don't want him to be traded," Mara continued. "I want him to be a Giant. I can't say it's a 100 percent guarantee it will happen. Can't say that about any player.''

Beckham has run into trouble after on-field tantrums and some off-field issues, such as the Miami trip with his wide receiver teammates the week before a January 2017 playoff game in Green Bay and a recent social media video that showed him and a woman in bed with a pizza, a brown cigarette and a powdery white substance.

But within the walls of the Giants' building, many say he's a good teammate and a hard worker. Gettleman acknowledged the latter at the end of Tuesday's interview.

"The kid works his fanny off," Gettleman said. "His [ankle] rehab is going terrific. He sends us pictures, sends us a video about him running. He's an excellent worker."

ESPN's Dan Graziano contributed to this report.