This paper considers bounded controllability for periodically switched linear systems in discrete time domain. Unlike the usual controllability problem, where the control values at each instant of time are unconstrained, we insist here that the control values at each instant of time belong to a pre-specified bounded set which contains zero in its interior. The concept of null controllability and asymptotic null controllability are introduced and necessary and sufficient conditions are established, respectively. The results are extended to the state delays case.