The 20-year-old midfielder is expected to link up with the Baggies on Thursday, Press Association Sport understands
Gnabry has recovered from a serious knee injury which forced him to miss all of last season and he has not played for Arsenal since March 2014.
The 20-year-old has made 18 appearances for the Gunners since joining from Stuttgart in 2011 and scored his only goal in their 2-1 win at Swansea in 2013
He made his Arsenal debut in a 6-1 League Cup win over Coventry in September 2012 and made his league bow in a 1-0 defeat to Norwich a month later.
Baggies boss Tony Pulis has been tracking the Germany Under-21 international as he looks to add to his midfield options ahead of Monday's Barclays Premier League opener with Manchester City
Gnabry will become Albion's fourth summer signing after Rickie Lambert, James McClean and James Chester with the club also keen on Cardiff goalkeeper David Marshall
The Baggies had also been in talks with Bakary Sako, after he lowered his wage demands, before the winger joined Crystal Palace on a free transfer on Wednesday.
Source : PA