Turnbull has a season left on his contract at Barnsley and has emerged as an option, with Whites head coach Uwe Rosler reportedly keen to sign a back-up keeper to potentially battle it out with the current number one, Marco Silvestri.
The 30-year-old has an impressive track record, playing for 10 clubs since 2002 and winning a Champions League winners medal with Chelsea in 2012, as an unused substitute in the final victory over Bayern Munich.
Turnbull is no stranger to Yorkshire, having made 27 starts in all competitions last term for Barnsley - keeping eight clean sheets - having also previously spent two loan spells with the Tykes earlier in his career.
He also had spells with Middlesbrough, Bradford and Doncaster.
The experienced shot-stopper would likely jump at the chance of joining Leeds, with the possibility of Championship football this term.