The Spain Under-19 international, who won the European championship in 2015, has agreed a three-year deal with the Premier League side and is reported to have cost $4.37 million.
Martinez will not officially join the club until 1 July but West Ham have confirmed he has been allowed to start training with his future team-mates in England.
"For me, it's a great opportunity to be here and I hope to make the most of being at such a big club," Martinez told West Ham's website.
"I joined West Ham because of the progress that the club is making. The decision to come here was made because it is a brilliant opportunity to join an interesting club in the Premier League."
Martinez, who has been compared to Atletico Madrid striker Fernando Torres, admits that the prospect of playing at the Olympic Stadium next season was a big factor behind his decision.
"The Premier League is well considered across the world and in Spain, we believe that it is one of the best leagues in the world," he said.
"This is a fantastic opportunity and I am fulfilling a lifelong dream by signing for one of the big clubs in the Premier League.
"The prospect of playing in front of 60,000 people next year really catches my attention, and it is a stadium that has so much history. I am hoping to be able to get the opportunity to do that and I hope that it will be soon."
