The Premiership club have announced the 32-year-old flanker will leave at the end of the campaign, with Wasps having already signed Brad Shields from the Hurricanes for next season.
"Hask goes back a long way with Wasps," said Wasps rugby director Dai Young.
"Since starting in the academy as a teenager he has played around 200 games including a spell as club captain, which is a great achievement.
"I have no doubt whatsoever, given his character and loyalty to this club, that he will remain fully committed until the end of the current campaign.
"On behalf of everyone at Wasps I'd like to thank James for his efforts and wish him well for the future."
Haskell is keen to compete in the Rugby World Cup 2019, but must play for an English club in order to be included in Eddie Jones' side.