Barnes helped Boston win the 2018 World Series, going 6-4 with a 3.65 ERA and 96 strikeouts in 62 games covering 61 2/3 innings as a hard-throwing setup man that season.
He had 24 saves in 2021, making the AL All-Star squad before struggling in the second half and losing the closer job. He fell to 0-4 with a 4.31 ERA and eight saves in 44 appearances last year, striking out 34 and walking 21 in 39 2/3 innings.
Bleier, 35, went 2-2 with a 3.55 ERA in 2022. He made one start and had a save.
Bleier, a Miami Beach native, made his big league debut with the New York Yankees in 2016. He is 14-6 with a 3.06 ERA in 308 career games, also playing for Baltimore.
Barnes, born and raised in nearby Connecticut, was drafted 19th overall by Boston in 2011 out of UConn. A converted starter, he is 32-30 with a 4.07 ERA and 47 saves in nine big league seasons.