The 19-year-old reportedly underwent a medical with the Hammers on Friday and has now tweeted using the 'Come on you Irons' hashtag.
He wrote: "Not ignoring everyone, crazy but exciting last few days. All will be revealed in time. Thanks for the messages #COYI."
Poyet, the son of Sunderland manager Gus, was named Charlton Athletic's Player of the Year last season despite not making his debut for the club until January.
He rejected a contract offer from the Addicks in order to make the switch to the Premier League.