The star defensive tackle and the Falcons have until July 15 to agree to a long-term contract. If a deal can't be reached, Jarrett will play the 2019 season on the tender, which is worth $15,209,000.
The Falcons view Jarrett, a 2015 fifth-round draft pick out of Clemson who turns 26 on Sunday, as the anchor along the defensive line.
He had more than 50 pressures last season and led the Falcons with 23 run stops, according to Pro Football Focus data. Jarrett had a career-high six sacks in 2018 to go with 52 combined tackles and three forced fumbles.
ESPN's Vaughn McClure contributed to this report.