He was given a non-objection certificate by the European Central Bank last week to play domestic cricket although he remains under suspension from the England team.
European Central Bank confirm aware Ben Stokes on his way to New Zealand to see family.
However, it is understood Stokes is heading to New Zealand to practice and spend time with his family, rather than meeting up with his England team-mates.
Stokes, who was born in New Zealand, was given permission to travel on Friday.
They would probably give Stokes 24 hours to recover from his flight before trying to contact him, Curwin said.
He said: "We've got a Canterbury Cricket board meeting tonight, so it will be discussed there". However, they have yet to receive a formal request.
Stokes was initially named in England's 16-man squad to tour Australia but was quickly stood down from global duty following his arrest on suspicion of causing actual bodily harm in Bristol in September.
Ben Stokes. Pic Getty
Canterbury coach Gary Stead said Stokes' possible involvement or not in the Ashes "hasn't been discussed whatsoever".
The England all-rounder was ruled out of his national side's opening Ashes Test with Australia due to an ongoing police investigation, relating to a street brawl in Bristol two months ago. "His travel arrangements have not been arranged by the European Central Bank".
After pictures of Stokes at Heathrow airport prompted speculation that the suspended all-rounder could be returning to England for the Ashes in Australia, it was confirmed that he was instead flying to New Zealand.
Stokes was arrested for his part in an incident outside a Bristol nightclub following England's third one-day worldwide victory against West Indies.
He was released by police while the investigation continues.
If he's playing cricket, why can't he just play for England?More news: Trump touts leadership during Thanksgiving message to troops