The Hammers have added to their strikeforce this summer as they struggled in front of goal last season, with Carroll scoring just twice in the Premier League.
However, Magpies boss Alan Pardew is hopeful that he can revive the England international's career and help him find the form that persuaded Liverpool to pay £35m for him in 2011.
Newcastle believe that a bid in the region of £15m may be enough to secure the 25-year-old's services from Sam Allardyce's side.
A source told The People: "Like all Geordies, Andy Carroll has never lost his love for Newcastle and we are sure if West Ham agree to sell we would have no difficulty in persuading him to come home."
Carroll moved to West Ham on a permanent basis last year in a £15m deal from Liverpool.