The move, which is intended to broaden the trio's footballing education, forms part of a relationship being struck up between Boro and Atletico.
It is hoped that both sides will benefit from the understanding, with players set to pass between the two.
Williams, Morris and Fewster are due in Spain on Thursday, offering them an opportunity to immerse themselves in a different culture.
It is hoped that they will return to England having taken plenty from the experience, with Boro's U18 and U21 squads also due to take in spells in Madrid facing their counterparts at Atletico.
Academy manager Dave Parnaby told the Evening Gazette: "We are keen to build relationships with continental clubs to expand our education.
"It's not just about Atletico Madrid playing Middlesbrough, it's designed to encourage player and coach development and foster positive public relations.
"It's a great opportunity for Bradley, Bryn and Luke to sample some different technical and tactical work and their spell over there will undoubtedly benefit them as individuals and players."