In a move White described as "disappointing", he said Wright will leave the Black Caps after the upcoming tour of the West Indies.
"John has been an outstanding servant for cricket in New Zealand over a long time and it is disappointing to lose someone of his calibre," White said in a statement.
"We were keen to see him continue his head coach role, however understand and respect his decision to look for another challenge."
A gritty former opening batsman and one-time India coach, Wright took over a Black Caps outfit in disarray and guided them to the semi-finals of last year's World Cup, as well as a rare Test win over Australia in Hobart.
The statement gave no reason for Wright's departure, although he has reportedly had a tense relationship with Australian John Buchanan, who was appointed NZC's director of cricket in April last year.
"I appreciate the offer to extend my contract but after much consideration I feel it's the right time to move on and look at other opportunities," Wright said in the statement.
No immediate replacement was announced.
"We are keen to fill this critical role as soon as possible but are conscious that we need to take the time to find the right candidate who can take us through to the 2015 ICC Cricket World Cup," White said.
New Zealand will tour the West Indies in July and early August.