The deal is pending a physical, per ESPN's Adam Schefter.
The Bengals seemingly confirmed reports of an agreement on Twitter.
Cincinnati drafted Burrow first overall in April after the signal-caller posted a historic senior campaign at LSU. Burrow completed 76.3% of his pass attempts for 5,671 yards and 60 touchdowns en route to a national title.
Burrow will take over as the Bengals' starting quarterback after the team released Andy Dalton - who was Cincinnati's No. 1 option under center since 2011 - this offseason.
Every top-five pick of the 2020 NFL Draft has now been signed.