The 23-year-old has struggled to make a telling impact since his £35 million switch from Newcastle United in January 2011 and new boss Brendan Rodgers is reportedly prepared to sanction a temporary switch as he tries to implement his passing game at Anfield and switch from a 4-4-2 approach to a more modern 4-3-3.
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The Hammers feel they hold an advantage over speculated interest from AC Milan given the friendship the striker shares with Kevin Nolan while the pair where at the Sports Direct Arena.
However, Carroll’s £80,000-a-week wages are believed to be a stumbling block for the London side.
Despite Carroll's failures at Anfield so far, an upturn in form near the end of the season - netting in both the FA Cup semi-final and final - saw him claim a spot in Roy Hodgson’s England squad heading to Euro 2012, where he scored against Sweden in the group stages.
Meanwhile, Rodgers is also hoping to tie up a deal for young Italian forward Fabio Borini from Roma, as well as leading the race for Daniel Sturridge’s signature, as exclusively revealed by Goal.com.
Carroll is said to be unsure on a move to Italy, but Liverpool would rather let the striker move abroad where he can not come back to haunt the club in the Premier League.