The 33-year-old's departure left Bahrain's preparations for January's Asian Cup in disarray and the Gulf state's football association said on Friday they would challenge his exit through the courts.
New Zealand Football had hoped to appoint Hudson as Ricki Herbert's successor last week but were forced to delay their announcement. They have called a news conference for Tuesday.
Herbert, who had eight years at the helm, stepped down after the 'All Whites' lost a two-legged playoff to Mexico 9-3 on aggregate and failed to reach this year's World Cup finals.
Unlike neighbours Australia, who moved to the Asian Football Confederation in 2006, New Zealand have stuck with the Oceania region, which offers easier opponents but no automatic berths for the World Cup finals.
Hudson, born in the United States when his former England midfielder father Alan was playing for the Seattle Sounders, was described as a "young Jose Mourinho" by Harry Redknapp after working under the now Queens Park Rangers boss at Tottenham.
He got his first head coaching job at senior level in 2011 with Welsh non-league club Newport County, where he lasted just five months before being sacked.
He moved to Bahrain to work with the under-23 side under fellow-Englishman Peter Taylor in 2012 and led the team to the title at the 2013 Gulf Cup of Nations.
After taking over the senior team in August 2013 he helped them qualify for the Asian Cup as group winners and signed a new contract through to 2016 earlier this year.
Adrian Whitbread, who had been coaching Bahrain's Olympic team, will take charge of the senior side for a two-week training camp in Spain later this month, the Bahrain FA said in a statement.
The 2015 Asian Cup takes place from Jan. 9-31 in Australia.
(Reporting by Nick Mulvenney, editing by Peter Rutherford)
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