"Such injuries take about three weeks to recover from. He is therefore out of Tuesday's last T20I match and the upcoming ODI series," he said.
The three-match T20I series is currently locked at 1-1, with Zimbabwe having won the opening match by 17 runs.
A BCB spokesman told AFP the captain for Tuesday's series decider would be announced in due course.
Nurul replaced batsman Mahmudullah Riyad as captain for the Zimbabwe T20Is, and is widely expected to lead Bangladesh in the upcoming T20I World Cup in Australia in October-November.
Bangladesh will play three ODIs against Zimbabwe on 5, 7 and 10 August, all of them at the Harare Sports Club.