Cumberland visited Chargers Park over the weekend.
Cumberland, 28, earned a roster spot with the New York Jets as an undrafted rookie free agent in 2010. He developed into a starter by 2012, a position he held until last season.
Kellen Davis supplanted Cumberland as the Jets' starting tight end last year. Cumberland played in only 258 offensive snaps (23 percent) for the Jets in 2015.
The Jets released Cumberland last month, saving $1.9 million in cap space. Cumberland totaled 86 catches, 1,119 yards and 10 touchdowns in six seasons with the Jets.
After losing Ladarius Green to the Pittsburgh Steelers in free agency, the Chargers were in need of experienced depth at tight end. San Diego re-signed veteran tight end Antonio Gates to a two-year, $12 million deal at the start of free agency.
However, Gates turns 36 years old in June. San Diego has three other developmental tight ends on the roster in Sean McGrath, Asante Cleveland and Tim Semisch. Those three -- combined -- have 27 catches in the NFL.
ESPN NFL Nation Jets beat writer Rich Cimini contributed to this report.