The France international is believed to be high on Brendan Rodgers's transfer wishlist, despite enduring another season blighted by injury.
He has made just 13 appearances this term, scoring three goals and making two assists.
Meanwhile, Carroll's future at Anfield appears uncertain just two years after swapping Tyneside for Merseyside in a deal worth £35m.
According to The Mirror, the Reds have made contact about the possibility of a swap deal involving the pair, which the Magpies are now considering.
Carroll has scored five goals in 19 appearances while on loan at West Ham United.
Rodgers has said that he will wait until the end of the season before discussing the striker's future.