The out-of-sorts opening batsman has a lung infection and selectors said Monday he will be given until Tuesday to prove himself strong enough to travel with the team.
Australia begin their tour with a Twenty20 match in Auckland on Thursday followed by a series of five One-day internationals and three Tests.
Hayden, 33, has been struggling since the tour of India last October and was recently diagnosed with pneumonitis, a lung inflammation related to pneumonia.
He has been resting at home and has been making progress, but his doctors do not yet believe he is well enough to guarantee he can get on the plane on Wednesday.
"Matthew's condition has improved and we will give him every chance to be fit by waiting until tomorrow to make any decision," chairman of selectors Trevor Hohns said Monday.
"He'll be examined by the doctors and necessary people tomorrow."
Hayden's Queensland teammates James Hopes and Andrew Symonds were both declared fit on Monday to travel to New Zealand after fitness tests.
Symonds had an Achilles tendon problem which forced him out of the second triangular series final against Pakistan in Sydney eight days ago.
Hopes, selected to make his Australian debut, has recovered from hamstring soreness.
Doubts over Hayden give Michael Clarke the chance to cement his spot as opener alongside Adam Gilchrist, leaving a place for someone further down the batting order.