Torres, 27, is a lefthander and had a 3.33 ERA and 51 strikeouts in 54 innings across 70 games last season for the Padres.
He started his big league career with the Tampa Bay Rays, but was dealt to the Padres before last season.
New York is expected to open the year with Torres joining Sean Gilmartin as the only lefthanders in the bullpen.
Mazzoni, 25, was the Mets' second round pick in the 2011 Draft. MLB.com had recently named him as the team's 16th best prospect heading into the 2015 season. He made 14 starts in 2014, split between four minor league stops and compiled a 7-2 record, 4.68 ERA, and 1.29 WHIP across 77.0 innings.
- Aaron Somers