Both players are free agents at the end of the season and have cleared revocable waivers.
Blevins, 34, was acquired by the Mets from the Nationals in 2015 for outfielder Matt den Dekker. He re-signed with the club after the 2015 season and then again after the 2017 season.
Since coming to New York, the left-hander has a 3.02 earned-run average with a 3.38 FIP, 159 strikeouts, 55 walks, and a 132 ERA+ in 209 appearances (131 innings).
This season, Blevins has a 3.86 ERA and a 96 ERA+ in 35 innings pitched (54 appearances).
Mesoraco, 30, was acquired from the Cincinnati Reds in the Matt Harvey trade in May.
The former first-round pick has had his career derailed by injuries but showed signs of life in 78 games this season, slashing .213/.321/.390.
We will keep you posted as new details emerge.