The 19-year-old, son of Sunderland manager Gus Poyet, is a free agent after his contract with the Addicks expired.
Poyet has reportedly also attracted interest from Chelsea, Manchester City and the Black Cats.
However, he has already had a medical with the Hammers - according to Sky sources - and now appears set to sign a deal at Upton Park.
He wrote on Twitter: "Not ignoring everyone, crazy but exciting last few days. All will be revealed in time. Thanks for the messages #COYI."
The initials at the end of his message stand for 'Come On You Irons', a reference to West Ham.
Poyet only broke into the Charlton team in January but his performances saw him named the Championship club's player of the year.
However, he turned down the chance to stay at The Valley and is now poised to follow in the footsteps of his father, who represented Chelsea and Tottenham, by playing in the Premier League.
Charlton will be due compensation for Poyet as he is the under age of 24, with a tribunal settling the fee.
Once the move is complete he will become West Ham boss Sam Allardyce's fourth summer signing, following Cheikhou Kayoute, Mauro Zarate and Aaron Cresswell.