They have offered to pay an initial £2m loan fee before making a permanent deal next summer.
Liverpool signed him from Newcastle in a record £35m deal under Kenny Dalglish in January 2011.
Carroll has endured a difficult time at Anfield since becoming the most expensive British footballer in history and he would like to return to Newcastle.
Sam Allardyce's West Ham failed in an original approach to sign the England international solely on loan earlier this summer but have returned with a permanent option.
Liverpool want £20m-plus.
Source: FOOTYMAD
