The likes of Blackburn Rovers and Ipswich Town have been linked with moves for Henderson, but Charlton are now believed to be leading the race for the Irishman.
Highly-rated Henderson has found himself down the pecking order at West Ham, with Spanish keeper Adrian and Jussi Jaaskelainen ahead of him in Sam Allardyce's plans.
Henderson, who has made just two appearances for West Ham, has impressed during loan stints at Ipswich and Bournemouth during the past two seasons.
Charlton are in the market for a new keeper after seeing Ben Hamer leave for Leicester and Henderson is thought to have emerged as their top target.
Should Charlton land Henderson his arrival will come as a boost to new boss Bob Peeters as he continues his plans ahead of the new season.