The striker has won plaudits around Upton Park for the way he has battled back from injury to lead the line so impressively in recent weeks for Sam Allardyce's men
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Carroll has impressed during his season-long stay at Upton Park, despite suffering niggling knee and hamstring injuries, and has convinced the likes of Sullivan that a deal would be in the club's best interests having fired seven goals for a club on course for a top-half finish.
Sullivan told reporters: "We would love to keep him beyond this season and we will soon be sitting down with Andy and Liverpool to make some progress on that front.
"Andy has battled back from injury twice to put in a series of tireless shifts for the team. His game is about so much more than just scoring goals.
"He is so often the focal point of everything good we do going forward."
Goal.com exclusively revealed that the 24-year-old will snub a move to the Hammers this summer, as he wants to stay and fight for his future on Merseyside but West Ham have apparently not been dissuaded from a move.