The Mackems, who alongside Bournemouth were listed as 6/1 outsiders earlier this week, are now clear frontrunners for Green's signature, after climbing above Middlesbrough and Arsenal [both 4/1].
Leeds United's Robert Green
Their sudden leap does have an air of inevitability about it, with Vito Mannone's imminent departure to Reading confirmed by Sunderland manager Simon Grayson on Tuesday.
And the Wearside club have already sold last season's first choice 'keeper Jordan Pickford to Everton.
Meanwhile, Leeds have recruited Felix Wiedwald from Werder Bremen, and the German is in line to replace Green as Thomas Christiansen's number one at Elland Road.
Green, 37, signed a new one-year contract at Leeds earlier this year, but would not have anticipated dropping out of contention.
Sunderland's Vito Mannone
Sunderland fielded youngster Maksymilian Stryjek in goal during Tuesday's friendly win against Hartlepool United, with last summer's recruit Mika their only other option in Mannone's absence.
Will Sunderland move for Leeds' Robert Green - and should they?