Wright, 21, had a trial with the Championship side Peterborough United but was not offered a contract.
His brother Mark, sister Jessica and friend Jack Tweed already appear in The Only Way Is Essex, the show is described as "real people in modified situations, saying unscripted lines but in a structured way".
According to a "source" speaking to Closer magazine: "Josh can be a bit of a bad boy and a loose cannon. He's been on lots of big nights out with Jack and has already been splashed across the papers larking about in women's underwear.
"He loves to party and has been told to 'ruffle a few feathers' on the show - there's no doubt he'll do that."
Wright faded badly from the first-team picture at Glanford Park last season, with a number of fans turning against him due to a perceived lack of effort.
The club could have ensured a transfer fee by offering him a contract extension, but chose not to to avoid running the risk he would accept it.