The 21-year-old has signed a four-year contract after passing a medical at The Hawthorns on Tuesday.
Boro will be due 15% of any sell-on fee.
Albion boss Tony Mowbray told the club's website: "James is a very talented young player and we're delighted he's agreed to join us
"He's a creative midfield player and will bring versatility to our attacking options.
"Despite the fact he's only 21, he already has more than 60 Premiership appearances under his belt and was a regular in the Boro side which reached the UEFA Cup final.
"We're now giving him a platform to be a part of something we're trying to build here.
"We're delighted he sees this as the place to progress his career and hopefully help us be successful."