Giacomini, 30, is in the third year of his four-year, $18 million contract that includes $7 million guaranteed. He stands to make base salaries of $4.5 million over the next two years.
In 2015, Giacomini appeared in all 16 games for the Jets and was rated by Pro Football Focus as the No. 66 offensive tackle out of 77 qualifying players.
Butler, 25, wound up signing on with the Ravens as an undrafted free agent out of Tennessee-Martin back in May of 2014. He managed to make the 53-man roster during his rookie season, but was waived and later signed to their practice squad last season.
The Ravens brought him back on an exclusive rights contract only to cut him loose coming out of the preseason. The Buccaneers signed him to their practice squad a few days later and promoted him to their active roster last week.
Tampa Bay cut Butler loose a few weeks ago and he was later claimed off of waivers by the Jets.
In 2015, Butler appeared in eight games for the Ravens and recorded 31 receptions for 363 yards receiving and no touchdowns.