Per Manish Mehta of the New York Daily News, Gase has responded to criticisms this year by telling people "I'm rich as f--k."
One Jets source told Mehta that Gase is "an insecure guy."
"In fact, some people on One Jets Drive are convinced Gase or a family member has a burner Twitter account to monitor what is being written or said about him—and defend him if necessary," Mehta wrote. "Gase didn’t respond to several requests from the News for this story."
Gase, along with Baylor's Matt Rhule and former Green Bay Packers head coach Mike McCarthy, were three finalists to take over the Jets after Todd Bowles was fired at the end of last season.
New York ultimately landed on Gase, who was fired by the Miami Dolphins following the 2018 season, as its head coach. The 41-year-old went 23-25 and made the playoffs once in three years with the Dolphins.
As the Jets prepare to wrap up a disappointing 2019 on Sunday, Gase has sounded indifferent at times to some of his team's struggles.
He told reporters on Oct. 7 "there’s nothing I look back on and say, 'I wish I would have done this over that'" following an 0-4 start in which the offense scored a total of two touchdowns.
Mehta noted there were two different occasions during the spring in which Gase "jogged off the field" when the defense was still practicing, and members of the Jets defense view him as a "glorified offensive coordinator."
Despite these issues, Jets CEO Christopher Johnson told reporters in November Gase will be back next season.
The Jets have won five of their past seven games following a 1-7 start. Their six wins this season are their most since a 10-6 record in 2015, and they haven't made the playoffs since 2010.