The injury-prone paceman was travelling to the Shere Bangla Stadium in Mirpur on Thursday morning when a bus shunted the rickshaw from behind, the Daily Star newspaper reported.
"The good news is that his injuries are not serious," Bangladesh team physiotherapist Baizidul Islam told the paper.
"But he will have to rest for now and cannot attend practice sessions."
The 31-year-old was not part of the test team to play India in Fatullah next week but was expected to lead the one-day side in the three-match series against the tourists in Dhaka later this month.
Bangladesh also lost Mahmudullah for the test after the all-rounder fractured a finger during practice on Thursday.
"Mahmudullah got his left thumb fractured during today's practice. He will miss the India test," team physician Debasish Chowdhury told reporters.
(Writing by Nick Mulvenney in Sydney, editing by Sudipto Ganguly)
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