McMillan played 55 Tests, 197 ODIs for New Zealand and has wealth of coaching experience.
The national team will have McMillan replacing former South Africa opener Neil McKenzie, who resigned last week due to the COVID-19 crisis.
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The BCB said that McMillan will join Bangladesh's pre-series training camp in Sri Lanka.
The 43-year-old played 55 Tests, 197 ODIs and 8 T20Is, scoring nearly 8,000 international runs before his retirement in 2007.
After his playing days, McMillan served as New Zealand's batting and fielding coach from 2014 to 2019. He also has experience of coaching Canterbury, Middlesex and Indian Premier League (IPL) team Kings XI Punjab (KXIP).
Bangladesh's three-Test series, part of the World Test Championship, is expected to kickstart in mid-October.