According to ESPN's Adam Schefter, an NFL executive said Mayfield is "definitely" in the conversation for Cleveland at No. 1 overall.
The Browns have long been in search of a franchise quarterback, and all signs point to them finally taking one with the top pick.
Cleveland could wait until pick No. 4 to take its signal-caller, but with the New York Giants and New York Jets each having a quarterback need at No. 2 and No. 3, respectively, the safe play would be to go with a quarterback at No. 1.
Mayfield is part of a quartet of options, as USC's Sam Darnold, UCLA's Josh Rosen and Wyoming's Josh Allen are also in the mix.
On Monday, ESPN's Todd McShay wrote that he has heard from multiple general managers that the Browns covet Allen with the top pick.
Allen has impressive physical tools, but his collegiate production was lacking. Mayfield is just the opposite.
Mayfield is probably the least physically impressive prospect of the top four quarterbacks in the class at 6'2" and 215 pounds, but he is coming off a Heisman Trophy-winning campaign.
As a senior with the Sooners in 2017, Mayfield completed 70.5 percent of his passes for 4,627 yards, 43 touchdowns and six interceptions.
If Cleveland passes on Mayfield at No. 1, there has been a lot of recent buzz regarding the Jets wanting him at No. 3.
Last week, Tony Pauline of DraftAnalyst.com reported that a source told him Mayfield to the Jets is a "done deal."
All four of the top quarterbacks in the draft appear likely to come off the board in the top 12 selections, and it is possible that five signal-callers could be gone by that point depending on how teams value Louisville's Lamar Jackson.
Bleacher Report's Matt Miller predicted in his most recent mock draft that the Browns will go with Darnold at No. 1, while Mayfield will fall to the Denver Broncos at No. 5 with a chance to learn behind Case Keenum.
Despite the fact that Cleveland is coming off an 0-16 season, it could be a good landing spot for Mayfield as well since there wouldn't be pressure to start immediately over veteran Tyrod Taylor.