The Kiwis will be able to begin their training after spending three days in quarantine and testing negative for the virus.
The Aussies flew into Bangladesh on a chartered flight last month and got on their buses from the tarmac after going through special immigration.
The New Zealand team, however, arrived in Bangladesh on a commercial flight and passed through regular immigration.
Two of their players, opener Allen and allrounder Colin de Grandhomme, had arrived in Bangladesh earlier after participating in the Hundred in England.
Allen tested positive for COVID-19 within 48 hours of arriving in Bangladesh.
In a media statement, Cricket New Zealand said Allen completed his vaccine dosage and tested negative in all tests prior to leaving England.
He is isolating at the team hotel and is showing moderate signs of the disease, the statement added.
The 22-year-old cricketer will be able to rejoin the squad after testing negative in tests carried out for two straight days after spending a particular number of days in isolation.
New Zealand’s last trip to Bangladesh dates back to the 2014 World Cup while the last time they played a bilateral series here was in 2013.
Bangladesh have made a slight change to the squad that triumphed over Australia while New Zealand lack any players from their World T20 squad.
The Tigers are yet to taste victory against the Kiwis in 10 games in the format. Bangladesh played New Zealand on their home turf only once in 2013, which ended in a defeat by 15 runs.
The series will kick off on Sep 1 and all the games will be held at Mirpur’s Sher-e-Bangla National Cricket Stadium.
New Zealand squad:
Tom Latham (capt, wk), Finn Allen, Hamish Bennett, Tom Blundell (wk), Doug Bracewell, Colin de Grandhomme, Jacob Duffy, Matt Henry (ODI only), Scott Kuggeleijn, Cole McConchie, Henry Nicholls, Ajaz Patel, Rachin Ravindra, Ben Sears (T20 only), Blair Tickner, Will Young.