It emerged that Mattock had previously handed in a transfer request last year, but stayed to help the Foxes win the League One title after lacking interest from other clubs.
However, after hearing of Di Matteo's interest this pre-season, Mattock handed in a second transfer request, and got his wish, with Nigel Pearson giving the Under-21 England international permission to talk to the West Midlands club.
The Baggies Boss said: 'Joe is exactly the kind of player we're looking for - a young, talented, English left-back who's got plenty of Championship experience despite his young years.
"He is hungry for success and he was very keen to join us. He's a quick, solid defender with a very good left foot.'
Speculation suggests that the deal for the Left-Back includes several add-ons and the Foxes could land up to £1.8m depending on appearances and promotion.
Meanwhile, Aman Verma's trial at Crewe Alexandra has earnt him a loan deal with the club.