Holders Toulouse are drawn with English Premiership sides Exeter Chiefs and Leicester Tigers, plus Irish province Ulster for the 2024-25 Investec Champions Cup.
England's Joe Marler and Will Stuart will start the first Test against the All Blacks in Dunedin on Saturday in two changes from the side that beat Japan a fortnight ago.
With Welsh woes in Australia going back to 1969, Warren Gatland's side face the Wallabies over two Tests and end the tour against Queensland Reds in Brisbane.