The injury will prevent Bashar from taking any further part in the Zimbabwe tour and has also ruled him out of playing in Kenya next month.
Bashar suffered a fracture on his left thumb when he attempted to catch opener Vusi Subanda at slip in the second one-day international against Zimbabwe in Harare yesterday.
Bangladesh beat Zimbabwe by 62 runs to level the five-match series 1-1.
Khaled Mashud will take over as stand-in captain for the three remaining matches against Zimbabwe and three One-dayers against Kenya.
Bashar will, however, remain with the team.