"For me, my job and my heart (are) here, doing everything I can to help the Cleveland Browns be good," McCown said Thursday at training camp.
McCown said since his family is from Texas, his phone has been "blowing up" with friends and family asking about the trade talk. Dallas needs a backup to Tony Romo after Kellen Moore suffered a fractured fibula Tuesday.
A league source told ESPN's Ed Werder that the Cowboys and Browns have discussed a potential trade for McCown, but Cleveland prefers to keep him, so they deliberately attached a high price to acquiring him. The Cowboys have rejected the proposal at this time. While an eventual deal still could happen, nothing is imminent, the source said.
In related news, Ron Freeman, the agent for his son Josh Freeman, told ESPN's Josina Anderson that he reached out to the Cowboys on Wednesday and was told team officials "are reviewing their options." However, on Thursday, a team source told Anderson that Dallas is "standing where we are until the right opportunity/player is available." Josh Freeman was on the Colts roster last season.
The free-agent quarterback is currently working out in Kansas City, where he has access to training partners and a suitable facility to stay ready for another opportunity in the NFL, the source said.
Cowboys executive vice president Stephen Jones has said that Dallas doesn't want to give up a quality draft pick for a backup and won't rush into a trade.
McCown, meanwhile, sounded like he's reluctant to be traded as well.
"Like I've told you guys before, you spend five minutes in this city and it's easy to be all-in," McCown said.
He also said the team addressed the trade talk with him, but would not detail what was said other than "we're on the same page."
Coach Hue Jackson said Thursday there is nothing to report.
"I have no thoughts whatsoever," Jackson said. "You said it, it's rumors. There's nothing to it as far as we're concerned. I've heard about it, obviously I have. But I feel good that he's here. He's a Cleveland Brown and I know nothing about the other part of it."
McCown's future has been a topic of discussion since the Browns signed Robert Griffin III in March. On Tuesday, Jackson said Griffin's progress was in part due to help from McCown, but the coach also said McCown being a mentor "has nothing to do with him remaining on the team."
McCown, 37, has been working with the second team behind Griffin during camp. Griffin, who has had ankle and knee injuries in his brief career, was healthy last season but did not play a down for Washington.
McCown started eight games for the Browns in 2015 but also had rib and collarbone injuries. He threw for 2,109 yards with 12 touchdowns and four interceptions.
"As a person of faith, I just say, 'Hey this is the day right in front of me,'" McCown said. "I'm going to live that out as best I can and move on to the next day. I just believe God's got the rest of that, and that will all sort itself out in the way it's supposed to."