Black Cats boss Roy Keane is determined to land his man and, after seeing an initial bid of £8million rejected, he has now reportedly upped his bid in his efforts to bring the Scotland star to the Stadium of Light.
Hearts were after closer to £10million for Gordon but the new package Sunderland have presented looks like being enough to wrest Gordon out of Tynecastle.
But both Sunderland and Hearts are refusing to comment on reports that a £9million deal is imminent.
But Keane - who is determined to establish Sunderland in the top flight - will not be afraid to bust the club's record transfer in order to get a player who is tipped to become one of the best in the world in his position.
Keane said only last week: "I would have no problem spending £8million or £10million on a player. It would not worry me."
And Gordon's agent Andy Evans insisted: "I'm not aware of any increased bid. He remains a Hearts player."
Gordon has two years left on his current contract and Hearts know his value will drop if he is allowed to go into the last season of his deal.
And should the Black Cats succeed in their efforts to sign Gordon, it would smash the existing British record transfer fee paid for a goalkeeper which was set by Manchester United's £7.8million capture of Fabien Barthez in 2000.
The deal would also be a record fee for any player leaving Scotland.
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