The Mets made the announcement Sunday and said both pitchers have been invited to big league camp.
The 34-year-old Hunter went 0-1 with one save and a 4.01 ERA in 24 games for Philadelphia last season. He also has played for Texas, Baltimore, the Chicago Cubs, Cleveland and Tampa Bay.
Hunter is 56-45 with 22 saves and a 4.08 ERA in 472 games during a 13-year career. A starter in his early days, he's pitched exclusively in relief since 2012.
The 31-year-old Montgomery pitched three games for Kansas City last year, giving up three earned runs in 5⅓ innings. Used in a variety of roles, he is 23-34 with three saves and a 3.84 ERA in 183 games over six seasons with Seattle, the Cubs and Kansas City.
Montgomery got a save in Game 7 of the 2016 World Series for the Cubs. He relieved with two outs in the 10th inning and a runner on first base and induced a groundout to preserve an 8-7 win at Cleveland.