Hull's Robert Snodgrass will meet West Ham on Friday to discuss personal terms and have a medical, Sky sources understand.
Snodgrass has chosen West Ham over Burnley after both clubs had £10m bids accepted by Hull.
The Scotland international has also attracted interest from Middlesbrough, whose £10m offer was structured differently to Burnley's improved bid.
Burnley manager Sean Dyche confirmed midfielder Dean Marney will miss the rest of the season with a knee injury and said the club are actively seeking signings in that area, but raised doubts as to Burnley's pulling power.
"Quite obviously, we're not the market leaders so we have to be aware of what might change in our own group," he said.
Middlesbrough boss Aitor Karanka also expressed concerns when confirming his interest in bringing in Snodgrass, suggesting the club's position in the Premier League could deter Hull from selling to a rival - with the two sides just four points apart.
Hull had activated an option to extend Snodgrass' contract in December, and his current deal is now due to expire in the summer of 2018.
The 29-year-old is Hull's top scorer this season, with nine goals in all competitions.