Greene was in his fifth season as New Jersey's captain.
The 37-year-old had spent all 14 of his NHL campaigns with the Devils, who signed him as an undrafted free agent out of the University of Miami (Ohio) in 2006.
Greene carries a $5-million cap hit. He's in the final season of the five-year, $25-million extension he inked with New Jersey in 2014.
Devils interim general manager Tom Fitzgerald said following the deal that the team's captaincy will remain vacant for the rest of the campaign.