Carroll has scored seven Premier League goals during his loan spell this season and Hammers co-owner David Sullivan is full of praise for a contribution that has lifted the club into the top 10.
"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," said Sullivan, quoted in the Daily Mirror.
"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."
The 24-year-old striker joined Liverpool in a £35million deal from Newcastle in January 2011 but has scored just six goals in 44 league games for the Reds.