Ian Rapoport of NFL Network provided the insight:
Joseph was the defensive coordinator under Gase when the two were in Miami in 2016. The Dolphins went 10-6 and made the playoffs before losing to the Pittsburgh Steelers in the Wild Card Round. Joseph then took the open Denver Broncos head-coaching gig and led the team to an 11-21 record before being relieved of his duties.
However, as Rapoport noted, Joseph could be heading to Cincinnati as the Bengals' next head coach. Albert Breer of Sports Illustrated wrote on Monday that the 46-year-old seemed "well-positioned" for the gig.
Katherine Terrell of ESPN also mentioned Joseph as a possibility:
If that potential opportunity falls through, then Joseph seemingly has a fallback option in New York. Rapoport wasn't the only reporter to connect Joseph and the Jets, as Rich Cimini of ESPN did as well:
A Gase-Joseph pairing may not go over so well with Jets fans, especially since the two were just fired at previous head-coaching stops with losing records.
Still, defense wasn't the primary problem for Joseph in Denver.
In 2017, the Broncos defense finished No. 10 in Football Outsiders DVOA (defense-adjusted value over average) before moving up to No. 5 this season. Granted, having superstars like Von Miller, Chris Harris Jr. and Bradley Chubb help, but Joseph deserves some credit for the defense's success.
He'll have some young talent to work with in New York (e.g. Jamal Adams and Darron Lee), so Jets fans could see some improvement on that end for a team that finished No. 29 in scoring defense.
Ultimately, a potential Gase-Joseph pairing isn't the flashiest coaching tandem, but it's hard to judge how the team will do before the introductory press conferences.