According to Cleveland.com's Mary Kay Cabot, Taylor has no issue with the No. 1 overall pick saying he wants to be the starter as a rookie: "He's a competitor, as well as myself. I think that's what drives us, that's what gets us to the point that we're at now in our career. Like I said, that doesn't faze me."
While Browns head coach Hue Jackson has been supportive of Taylor as the current starter, he said that Mayfield is the "future of this organization."
Both signal-callers will look to help the Browns rebound from a historically bad 0-16 season.
Jackson declared Taylor the starter after the Browns acquired him in an offseason trade with the Buffalo Bills, and he reiterated the stance even after drafting Mayfield, per Cabot: "I'm not going to back off of this. We can keep writing this narrative, Tyrod Taylor's the starting quarterback of this football team, and that won't change."
Taylor began his career as a backup with the Baltimore Ravens before starting for three seasons in Buffalo.
The 28-year-old went 22-20 as a starter, made one Pro Bowl and helped end the NFL's longest playoff drought by leading the Bills to the playoffs last season.
Taylor is coming off a 2018 campaign that saw him throw for 2,799 yards, 14 touchdowns and four interceptions while rushing for 427 yards and four scores in 14 games.
Meanwhile, Mayfield won the Heisman Trophy at Oklahoma by throwing for 4,627 yards, 43 touchdowns and six picks.
Although the Browns didn't win a game last season, it's fair to say that Taylor will have more offensive talent around him in 2018 than he did last season.
In addition to holdover wide receiver Josh Gordon, the Browns acquired wideout Jarvis Landry in a trade with the Miami Dolphins, signed running back Carlos Hyde in free agency and selected Georgia running back Nick Chubb in the second round of the 2018 NFL draft.
There is little doubt that the Browns took Mayfield No. 1 overall with an eye toward making him the starter in the near future, but based on the success Taylor enjoyed in Buffalo, Jackson and Co. have the luxury of grooming Mayfield until he is completely ready to take the reins.