Recent reports, such as those in the Yorkshire Evening Post, have suggested that Leeds were keen on a move for former Middlesbrough man Turnbull as they look to bring in another goalkeeper this summer.
Leeds manager Uwe Rosler is keen to bring in an experienced custodian in the transfer window as he wants to have cover and competition for No.1 Marco Silvestri next season.
Charlie Horton has already been drafted in from Cardiff City, but the American is seen as one for the future at Elland Road.
League One outfit Barnsley are thought to be willing to let Turnbull, who is entering the final year of his contract at Oakwell, move on as they look to trim their wage bill.
However, although 30-year-old Turnbull was on their list of targets, Leeds have now shelved their interest and have switched their attention elsewhere as they look to bring in goalkeeping reinforcements.