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Kane Williamson's side were due to arrive in August for the two-match series as part of the ICC World Test Championship.
The BCB and New Zealand Cricket would now work on identifying a suitable time to reschedule the series, the Bangladesh board said in a statement.
"In the present COVID-19 pandemic scenario, hosting a full cricket series in August 2020 will be a challenge in terms of preparations and we cannot take chances with the safety and health of players, support staff and related stakeholders," BCB chief executive Nizamuddin Chowdhury said.
"Under these circumstances, the BCB and NZC have felt that the best way forward would be to defer the series."
Bangladesh has reported more than 115,000 infections and 1,500 deaths due to the novel coronavirus.
Former captain Mashrafe Mortaza is among those to have tested positive while spinner Nazmul Islam and opener Nafees Iqbal are also reported to have returned positive tests.