Details of the contracts weren't immediately available.
The Jets actually pursued Beachum last offseason, but he wound up signing with the Jacksonville Jaguars. The Jets are overhauling their offensive line, and they see Beachum as their starting left tackle.
"Just a blessing to keep playing football, a blessing to be at a great organization," Beachum said in a Fox ProCast video from the Jets' facility. "J-E-T-S, Jets, Jets, Jets. I already know the chant, baby."
On Thursday, the Jets re-signed Ben Ijalana, who started the final eight games at left tackle, but the plan is to have him compete at right tackle with second-year player Brandon Shell.
The Jets also re-signed Tanner Purdum, their long-snapper since 2010, on Friday.
In recent weeks, the Jets released three starters along the offensive line: center Nick Mangold, OT Breno Giacomini and OT Ryan Clady. Until Friday, the team had been cutting players, and not signing anyone, causing the fan base to grumble.
The Jaguars declined the option in Beachum's contract in February, making him a free agent.
Beachum, 27, signed a one-year deal with a four-year option with the Jaguars last March and earned $5 million in 2016. Had the Jaguars picked up the option, Beachum would have had his $7.5 million base salary in 2017 and $5.5 million of his $8.5 million base salary in 2018 guaranteed.
Beachum returned from a torn left ACL suffered in October 2015 while with the Steelers and started 15 games last season in Jacksonville, missing only one game because of a concussion. He was a key part of an offensive line that gave up only 34 sacks, the fewest allowed by the Jaguars since 2007 (31).
Beachum dealt with soreness and slight swelling in his knee throughout the season, but that was expected as he returned to the field less than a year removed from his knee surgery.
Catanzaro, 26, spent his first three seasons with the Arizona Cardinals. He made only 75 percent of his field goal tries last season, prompting the Cards to let him test the market. The Jets envision him as the replacement for longtime kicker Nick Folk, who was waived last month.
ESPN's Michael DiRocco contributed to this report.