The 18-year-old, who is the son of Sunderland boss Gus, is highly-rated despite making only a handful of appearances for the Addicks.
Poyet, who has played just seven times for the Championship side, has attracted the attention of Hammers boss Sam Allardyce, according to talkSPORT.
Tottenham and Southampton are also thought to be keeping an eye on the situation.
Diego Poyet: "There were rumours of me moving to Sunderland, but for the time being I'd rather do it for myself. I've always said I wouldn't want to play under him.
"My dad is my biggest critic. It has disadvantages, but it has advantages because he has the knowledge and I can learn from him. We used to have a kickabout, but not all the time.
"He didn't want to pressurise me or push me when I was young into doing something. He just let me get on with it."