Rodriguez, 27, had been eligible for arbitration. Tuesday is the deadline for teams to tender contracts to arbitration-eligible players.
He did not pitch last season after testing positive for COVID-19 in July and later developing myocarditis, a heart condition associated with the disease.
In 2019, Rodriguez went 19-6 with a 3.81 ERA in 34 starts.
He is scheduled to become a free agent after the 2021 season.
USA Today Sports first reported news of the deal.
ESPN's Buster Olney contributed to this report.