The Jets waived kicker Nick Rose in a corresponding move.
Garcia, the Patriots' 2017 third-round pick, was waived by the team on Friday with a non-football-illness designation.
The Jets don't have a short-term need at left tackle -- Kelvin Beachum is the returning starter -- but they see Garcia as a player with potential to develop over time. If he makes it to training camp and performs well, he could challenge longtime backup Ben Ijalana for the No. 2 job. Ijalana re-signed in the offseason for one year, $2.4 million, but only $500,000 is guaranteed. The Jets didn't draft any offensive linemen.
The 6-foot-7, 302-pound Garcia was selected No. 85 overall last year after he was credited with 70 knockdown blocks and didn't allow a sack in more than 900 offensive snaps as a senior at Troy.
With Nate Solder entering the final year of his contract in 2017, the Patriots had traded a third-round pick (No. 96) and fourth-rounder (No. 124) to move up to select Garcia, with the hope of grooming him as a possible replacement in 2018. Solder signed with the New York Giants as a free agent this offseason.
But after taking part in the first few practices of 2017 training camp, Garcia never took the field again, ultimately being placed on the non-football-illness list. The Boston Herald reported that Garcia had blood clots in his lungs.
Garcia had been taking part in the Patriots' voluntary offseason program, which just concluded its fourth week.
ESPN's Rich Cimini and Mike Reiss contributed to this report.