The contract also includes an invitation to spring training, according to the reports.
It was originially reported that Reyes signed with the Philadelphia Phillies, but that deal was never finalized.
Reyes, 33, had a 3.55 ERA over 59 appearances with the Cardinals last season.
Reyes would earn $900,000 for making the team, with another $500,000 available through incentives, according to the reports.
In his career, Reyes has held lefties to a .238 average and .669 OPS. He experienced a reversal in 2010, however, holding righties to a .177 mark with a .481 OPS, while lefties hit .307/.862 against him.