According to Andrew Brandt of The MMQB, Mayfield's agent, Jack Mills, said: "We knew that the [New York] Jets at [No.] 3 was the bottom line. We had another team that said, 'You may get a big surprise on draft day at No. 2 if he is available.' And that team was the Patriots."
Cleveland nabbed Mayfield with the No. 1 overall pick, while the New York Giants stayed at No. 2 and took Penn State running back Saquon Barkley.
The MMQB's Albert Breer and Robert Klemko added that the Pats were denied a meeting with Mayfield until after they acquired the No. 23 overall pick as part of the trade that sent wide receiver Brandin Cooks to the Los Angeles Rams.
New England entered the draft owning selection Nos. 23 and 31, but it likely would have needed to offer much more to get to No. 2, including next year's first-round pick.
On Monday, Patriots quarterback Tom Brady officially committed to play in 2018.
Brady was the NFL MVP last season and turned in a dominant performance in New England's Super Bowl LII loss to the Philadelphia Eagles with 505 yards passing and three touchdowns, but he will be 41 by the time the 2018 season begins.
New England's primary backup is veteran journeyman Brian Hoyer, although New England drafted LSU signal-caller Danny Etling in the seventh round.
With their pair of first-round selections, the Pats took Georgia offensive lineman Isaiah Wynn and Georgia running back Sony Michel.