Simmons has been handed a short-term contract until next year's Champions Trophy, the Bangladesh Cricket Board (BCB) said.
The 61-year-old replaces Chandika Hathurusingha, who was suspended on disciplinary grounds, ESPNCricinfo reported earlier on Tuesday.
"Hathurusingha has two counts of misconduct," the website quoted BCB president Faruque Ahmed as saying. "First is about an assault on a player. Secondly, he took too many leaves, more than what was in his contract."
The BCB did not confirm that Hathurusingha had been sacked, but said Simmons would arrive in Dhaka on Wednesday and help the team prepare for their upcoming home Test series against South Africa.
"I have had the opportunity to speak to Phil Simmons and have listened to his cricket philosophy and ideas. His extensive coaching experience, vision and proven track record of success make him an ideal choice for this role," Ahmed said in the statement.
Hathurusingha, who coached Bangladesh from 2014-2017, returned for his second stint in January last year but Ahmed questioned BCB's decision to re-appoint the 56-year-old.
Ahmed took charge of the BCB in August following Nazmul Hasan Papon's resignation amid political upheaval in the country and said Hathurusingha should not continue as head coach.
Under Hathurusingha, Bangladesh failed to get past the group stage in the last two white-ball World Cups but registered a historic 2-0 Test series win over Pakistan in September -- their first overseas series victory in the format in 15 years.
However, India then whitewashed Bangladesh at home in a two-match Test series and secured a clean sweep against captain Najmul Hossain Shanto's side in a three-match Twenty20 international series last week.
Simmons, who was in charge when West Indies became T20 world champions in 2016, started his international coaching career with Zimbabwe in 2004 and coached several other teams such as Ireland and Afghanistan
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