The 27-year-old scored 14 times for Gillingham last season, but was released by the Gills earlier this week after a falling-out with head coach Ady Pennock.
Wright's roots are in Essex, and Shrimpers boss Brown has confirmed that he is in the hunt for the midfielder, who has also represented Charlton Athletic, Scunthorpe United, Millwall and Leyton Orient during a well-travelled career.
"I'm very interested in Josh," Brown told the Southend Echo. "He got something like 14 goals last season and goalscoring midfielders are a rarity in any league.
Wright took to Twitter earlier this week to deny suggestions that he refused to sit on the bench for Gillingham's League One match with Southend last weekend.