Warburton, 27, has been sidelined with a shoulder injury since 30 April and missed Sunday's 27-13 loss to England.
Prop Paul James will miss the first Test match in Auckland with a calf injury, but coach Warren Gatland hopes he will face the Chiefs on 14 June.
Second rower Alun Wyn Jones is in line to win his 100th cap for Wales in the game at Eden Park.
Wales have not beaten the All Blacks in 26 matches since 1953.
The match will be New Zealand's first since beating Australia 34-17 in the 2015 World Cup final.
Lock Luke Charteris will join up with the Wales squad on Saturday after being released by French side Racing 92.
Charteris and Clermont Auvergne centre Jonathan Davies both missed the defeat at Twickenham because they were on club duty, though Davies has already joined the Wales camp in New Zealand.
Hooker Ken Owens will also be ready for the game after overcoming a back problem, but blindside flanker Dan Lydiate has been ruled out of the tour because of a shoulder injury.
He has been replaced by Cardiff Blues open-side Ellis Jenkins, who is 23 and uncapped.
Gatland described Jenkins, a former captain of Wales Under-20s, as a "young, exciting player" with "pedigree".
He added: "He is a real intelligent rugby player, he's had a very good season, so it's nice for him to have the opportunity to come out here and hopefully gain some experience and maybe get some game time, too.
"If he's not involved in that first game, then hopefully he will get a chance at Hamilton against the Chiefs."
New Zealand lock Sam Whitelock is a doubt for the first Test as he recovers from a hamstring injury.
The All Blacks are already without experienced campaigners Richie McCaw, Dan Carter, Ma'a Nonu and Conrad Smith who have quit international rugby.
Wales captain Sam Warburton is expected to be fit enough to play against world champions New Zealand on 11 June.