The Hammers have already been busy in the transfer window, recruiting Razvan Rat and Adrian, and hope to add the Liverpool frontman to their squad after a successful loan spell
The Hammers have already signed defender Razvan Rat from Shakhtar Donetsk along with goalkeeper Adrian from Spanish side Real Betis and the 58-year-old is hopeful of more incomings - including Carroll.
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But, as reported by Goal in May, West Ham's hopes of siging Carroll were dented by his desire to prove himself at Anfield, though he remains top of Allardyce's summer window wishlist.
"At this moment in time, our first priority is signing Andy Carroll, if we can," Allardyce told the club’s official website.
"The work never really stops over the summer, but I can do it from where the sun is shining because of the communication levels you can achieve today with the technology.
"Via Skype or phone or email or conference call, it is pretty easy to keep in touch with everybody and we can all link-up together if need be.
"If there is an imminent signing, then I’ll be straight back to meet and greet the player and shake hands when he’s signed the contract and to welcome him to West Ham United."