The out-half, capped 16 times by the All Blacks, announced the news on his Instagram page and will sign a three-year deal with the Aviva Premiership side, according to New Zealand Rugby's official site.
Lima Sopoaga celebrates a try against the Springboks during the 2017 season.
Sopoaga's arrival is a strong sign that Wasps are happy to let Danny Cipriani leave when his contract expires at the end of the season.
Sopoaga, who recently became a father for the first time, said it was a hard decision to make.
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"I'd like to thank the Highlanders, All Blacks, Southland and Wellington for the awesome experiences you have provided me with over the years. I've made life long friends and memories that I'll cherish forever".
All Blacks coach Steve Hansen said he was disappointed but understood Sopoaga's reasons to begin a new chapter of his career in England.
Lima Sopoaga and Beauden Barrett of the All Blacks show off the Bledisloe Cup in August previous year.
Dai Young's side has undoubtedly got a seriously talented player on their hands with the 26-year-old looking perfectly comfortable in an All Blacks jersey since he made his debut against South Africa in the Rugby Championship 2015.