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Leicester had a Stg500,000 bid for the 25-year-old accepted on Monday morning and Connolly’s advisors travelled to the Walkers Stadium for talks but former Ireland manager Mick McCarthy has now decided to move for the striker.
Sunderland have not disclosed how big their bid is though chairman Bob Murray revealed it was greater than Leicester’s offer.
"David Connolly is a player we are very interested in. We made an offer to West Ham on Monday while the player was at Leicester City having his medical,” said Sullivan.
"West Ham accepted our bid, as it was a superior offer, and we are now in discussions with the player and his representative.
"The final decision lies with David. We all recognise the need for an additional striker and David has played for Mick McCarthy at international level and alongside both Gary Breen and Colin Healy."
Connolly joined West Ham for a bargain Stg285,000 last season from Milton Keynes Dons where he scored 42 goals in just 63 appearances proving himself to be one of the deadliest marksmen outside the Premiership.
If Sunderland did manage to claw away Connolly from Leicester Citys' grasp it would be a double blow for the Foxes as in the last week boss Mickey Adams has failed also to strike a deal with former Sunderland winger Jason McAteer about a move to the Walkers Stadium.