The 27-year old Australian international has been out of contract since he was released by Blackpool after they were relegated from the Premier League last season and now could find himself in a Hammers shirt if all goes well.
Carney played for a Hammers development squad against a Tottenham XI on Monday night and reportedly did very well - his performance apparently impressing West Ham boss Sam Allardyce.
A match report from West Ham read: "Carney has improved as the game has gone on and also caught the eye at the attacking end of the field.
"He's a strong player and experienced and composed in possession."
Carney was believed to be close to a move to Nottingham Forest during the recent transfer window but the deal fell through.