After catching a career-low 18 passes for 132 yards and three touchdowns in eight games last season with the Packers, Franks is expected to be a backup behind Chris Baker this season.
Franks, who spent the last eight seasons with Green Bay, was cut last month. He was due to make $3 million in base salary.
According to several New York-area media reports, the Jets had originally hoped to sign tight end Ben Utecht. When that plan fell through after Utecht signed an offer sheet with the Bengals, New York focused its efforts on Franks.
Between 2001 and 2004, Franks caught 27 touchdowns. He made the Pro Bowl in the 2001, '02 and '03 seasons. Injuries have slowed Franks in recent years after he played only 10 games in 2005 and eight games in '07.