Jones finished with a 17-19 record at Duke before the Giants made him the No. 6 overall pick.
Mayfield wasn't the consensus No. 1 overall quarterback prospect before the 2018 draft despite going 34-6 at Oklahoma. But he was at the top of the Browns' draft chart as they selected him No. 1 overall and watched him lead the franchise to a 7-8-1 record after the team went 0-16 the previous season.
"Either you have a history of winning and being that guy for your team or you don't," he told GQ.
Later on Tuesday, Mayfield clarified his comments in a post to his Instagram Story, writing: "This is not what I said ... just so we're clear. I also said I was surprised I got drafted number one. Then was talking about the flaws in evaluating QB's. Where I brought up winning being important. Reporters and media will do anything to come up with a click bait story. Heard nothing but good things and wish nothing but the best for Daniel."
Asked Tuesday about Mayfield's comments, Jones said, "I'm not sure I would do it that way, but everyone's got a different way of doing things.
"My focus is here, our focus is here. I think I speak for the team. We're very focused on what we're doing here and getting ready to play going into the season," he said.
Jones, who said he thinks Mayfield is a "great player," said he wished he had won more at Duke.
"I wish we did win more games, and I played a big part in that. I take responsibility for that," he said.
Asked if he thinks he's a winner, Jones said, "I do."
"I've always felt like I am. Growing up competing. I think I'm a competitor. I think I'm a winner," he said.
Having a losing record in college doesn't mean a quarterback won't be successful in the NFL. Kansas City Chiefs quarterback Patrick Mahomes, the reigning NFL MVP, was 13-16 at Texas Tech. Former Denver Broncos quarterback John Elway went on to have a Hall of Fame career, winning an NFL MVP and two Super Bowls, after going 15-18 at Stanford.
Mayfield covered several other topics in the interview, including when he planted the Oklahoma flag at midfield in Ohio Stadium after the Sooners defeated Ohio State in 2017. Mayfield said that the "higher-ups" at OU asking him to apologize for it was "just jaw-dropping" to him.
"Actually we won. That's what we're about. I had done so much and worked so hard to play for that school, I was just kinda ... almost embarrassed for them to tell me to apologize," said Mayfield, who, when asked how heartfelt that apology actually was on a scale of 1 to 10, replied, "zero."
"Which might hurt some Ohio fans' feelings," he added. "But I think we're all good now."
Mayfield wasn't the only Browns player to criticize the Giants in a story published Tuesday.
Wide receiver Odell Beckham Jr., who has been outspoken about the Giants since they traded him to the Browns in March, told Sports Illustrated that he believes the trade was "personal." Beckham told the magazine that he believes that the Giants had better offers from other teams but traded him to the Browns hoping to stifle his career.
"This wasn't no business move. This was personal. They thought they'd send me here to die," he said.
The Giants declined to comment when contacted by SI about Beckham's remarks.
Browns coach Freddie Kitchens was asked Tuesday if he's worried about the bull's-eye on his team getting bigger because of the talk coming from his players.
"We don't care. We don't care. It's already on there. We'll be ready to play. I don't know what a bull's-eye is. ... If they're not trying to beat our ass and we're not trying to beat their ass, I don't know what else you do. Because that's what we're going to try to do. Hopefully they try to do the same," he said.