The Browns are coming off a winless season in 2017 and are 1-31 during their last two years.
Tebow, however, credited the Browns with "making some moves to give themselves a chance," namely selecting Mayfield with the top overall pick of the 2018 NFL Draft.
"I'm friends with (Mayfield) and I think he's going to do a good job," the 30-year-old Tebow said, per Cleveland.com. "I think he's really focused on trying to prove that he was worth it."
Tebow, who currently plays for the Double-A Binghamton Rumble Ponies, won the Heisman Trophy in 2007 with Florida. Mayfield captured the trophy 10 years later with Oklahoma.
Mayfield will begin his rookie season behind Tyrod Taylor on the depth chart for the Browns, who open training camp on July 26.
The 23-year-old Mayfield passed for 4,627 yards and accounted for 48 touchdowns with just six interceptions while leading the Sooners to the College Football Playoff last season.
Taylor played in 15 games for the Buffalo Bills and completed 62.6 percent of his passes for 2,799 yards and 14 touchdowns with four interceptions last season.
The 28-year-old Taylor guided Buffalo to the postseason for the first time since 1999 last season before being sent to Cleveland. Now, he's aiming to have the Browns reach the playoffs for the first time since 2002.
The 25th overall pick of the 2010 NFL Draft by the Broncos, Tebow led Denver to an 8-6 record in 14 starts in 2010 and 2011. He spent 2012 as a reserve with the New York Jets, his last regular-season NFL action.