The Express and Star report states that the youngster will link up with the Baggies academy this season, and should have arrived at the club for training at the start of the week.
He is pictured holding the West Brom shirt on his Instagram account, confirming an official move.
House made several appearances for the Gunners' youth teams last season, as well as being named on the bench at both Under-18 and Under-19, with his most high profile appearances being a 4-3 loss away to Anderlecht in the Uefa Youth League.
But he was not offered a new deal, and has now moved to the Midlands to continue his footballing career.
Why West Brom could be a good place for House to develop
Despite his release by Arsenal, the Baggies could be an excellent place for House to try and improve.
Tony Pulis has a selection of British 'keepers at his disposal that could prove to be excellent role models for the youngster, including England man Ben Foster and former Wales international Boaz Myhill.
House also has a better shot of edging closer towards the first team at the Hawthorns, with Arsenal set to add to their goalkeeping strength this week with the expected signing of Petr Cech.
With Foster and Myhill both in their 30s, there is an opportunity for a young stopper to make a play for a first team role of some description in the next few years, if they can really impress the former Stoke boss.
With it difficult to judge a goalkeeper's qualities at such a young age, House could well flourish in his mid to late teenage years and leave the Baggies very happy with their deal.