The 26-year-old has struggled for fitness and form since leaving Newcastle for Liverpool in 2011, when he became the most expensive player in British history, costing £35million.
His spell on Merseyside was short-lived as he moved to West Ham in 2012, firstly on loan and then on a permanent deal, but he has started fewer than 50 league games for The Hammers in over three years.
Now their patience looks to have worn out and HITC Sport understands that they want to move him on, and the prime candidates to take him are Newcastle - who have never lost an interest in their former marksman.
HITC Sport has learnt that Newcastle will take on his near £80,000-a-week wages but they will not be willing to pay an excessive transfer fee and nowhere near the report £10million plus that West Ham are said to be looking for.
Newcastle are very much looking for a new striker in January, but despite their interest in Carroll, they are still in the hunt for West Bromwich Albion star Saido Berahino, who they and Tottenham Hotspur both tried to land in the summer.
Should Newcastle land their striker target then that will likely signal the departure of Papiss Cisse, who looks set for a move to the Middle East.