England Under-20 international Bassett was the Championship's top finisher last season with 14 tries. He also played for the RFU Championship XV against New Zealand Maori five months ago.
Wasps confirmed 21-year-old Bassett's move to Adams Park, where he will join fellow wings Tom Varndell and Christian Wade, who have claimed 22 Premiership touchdowns between them this term.
"Josh fits the bill perfectly for what we are trying to achieve here," Wasps rugby director David Young told the club's official website.
"We have some very talented wingers in our squad, but he's got a big future, and having impressed hugely at Championship level we are looking forward to seeing what he can do with us."