Kiper also mocked Allen to the Browns in his first mock draft on Jan. 18.
Cleveland owns the No. 1 and No. 4 overall picks this year, and it remains in search of an elusive franchise quarterback.
In his write-up, Kiper was confident the Browns will take a quarterback first as long as they don't sign Kirk Cousins in free agency.
Kiper also compared Allen to Detroit Lions quarterback Matthew Stafford, who has enjoyed a productive NFL career and led his team to the postseason three times.
Allen's talent is undeniable, but he is coming off a rough season in which he completed only 56.3 percent of his passes for 1,812 yards, 16 touchdowns and six interceptions.
It was a notable drop-off from 2016, when he threw for 3,203 yards, 28 touchdowns and 15 picks.
Allen has impressive athleticism, though, and prototypical size at 6'5" and 233 pounds.
Which Teams Are the Best Fit for Dez?
Johnny Football Is BACK!
Quavo's Mega-Celebrity Flag Football Game
Meet the NFL's Next Alshon Jeffery
Tremaine Edmunds Is NFL Draft No-Brainer
Grading the Rams' Deal with Suh
Meet the Undisputed Top CB in the NFL Draft
Grading Jordy Nelson's Deal with the Raiders
Grading the Kirk Cousins Contract
Grading the Allen Robinson Contract
Guice Is the Superstar Prospect to Remember
Simms: Lamar Jackson Is No.1 QB in 2018 Draft
Biggest Winners and Losers from NFL Combine
Where Will Jimmy Graham Land?
Simms: Barkley Is on a Different Planet
Where Would the Dolphins Trade Jarvis Landry?
Where Will Kirk Cousins Land?
Building the Perfect QB Prospect for the Draft
James Washington Is the Top Riser at WR for NFL Draft
He has no shortage of competition when it comes to being the first quarterback selected in April's draft in Dallas. USC's Sam Darnold, UCLA's Josh Rosen, Oklahoma's Baker Mayfield and Louisville's Lamar Jackson are in the mix as well.
In his most recent big board, Bleacher Report's Matt Miller ranked Allen as the No. 7 overall player in the draft and the No. 3 quarterback behind Darnold and Rosen, respectively.
Also, Miller predicted in his latest mock draft that the Denver Broncos will take Allen fifth overall.
If the Browns strike first and take him No. 1, they will look to Allen to end the NFL's longest playoff drought, as they haven't reached the postseason since 2002.
Second-round pick DeShone Kizer struggled mightily as a rookie quarterback with the Browns last season. However, pairing Allen with a veteran free agent such as Sam Bradford or even AJ McCarron, who backed up Pro Bowler Andy Dalton for four seasons in Cincinnati, could go a long way toward preparing him for life as an NFL starter should Cleveland opt to go that route.