Gase was sacked at the end of the season by the Miami Dolphins after three seasons.
The Dolphins went 23-25 during Gase's tenure.
McCarthy was sacked as the Packers' head coach mid-way through the 2018 season.
Adam Schefter and Jeff Darlington of ESPN first reported the news.
Gase followed coach John Fox to Chicago in 2015 to be the Bears' offensive coordinator and was hired by the Dolphins as their head coach the following season. Miami fired him following the end of the 2018 regular season.
This season, the Dolphins finished 7-9 and fired Gase the day after the last game.
Miami faces the Jets twice a year, as both are in the AFC East. Playing in Gase's Broncos offense in 2013, quarterback Peyton Manning enjoyed a record-breaking season, passing for 5,477 yards and 55 touchdowns en route to a Super Bowl.
More than anything else, the Jets were looking for their next coach to be able to develop Sam Darnold into their franchise quarterback.
National Football League head coaching vacancies have started to get filled up, but Mike McCarthy remains on the market. However, Rhule reportedly pulled out of contention for the job after being told he wouldn't be given control to hire his own assistants, per SNY's Ralph Vacchiano.