The out-of-contract midfielder has been courted by the Irons for several weeks and the £50k a week offer would blow the Black Cats hopes of keeping him out of the water.
And they claim too that Newcastle United – who are keen – would not be able to rival the Hammers bid.
Irons boss Sam Allardyce has identified Sunderland’s Academy product Colback as the best value of the Bosman midfielders around.
And with no transfer fee involved wages of £2.5 million a year are within the club’s parameters and seen as justifiable for a player who also interested Everton last season.
Newcastle have been monitoring Colback’s situation closely but have yet to make an offer and are not expected to do so now in light of the West Ham development.
The Echo declares: “The only hope of Colback staying in the North East is if the player himself and his young family decide to stay in the region.”