Summary: The IDR$(s)$ based on the induced dimension reduction (IDR) theorem, is a new class of efficient algorithms for large nonsymmetric linear systems. IDR(1) is mathematically equivalent to BiCGStab at the even IDR(1) residuals, and IDR$(s)$ with $s>1$ is competitive with most Bi-CG based methods. For these reasons, we extend the IDR$(s)$ to solve large nonsymmetric linear systems with multiple right-hand sides. In this paper, a variant of the IDR theorem is given at first, then the block IDR$(s)$, an extension of IDR$(s)$ based on the variant IDR$(s)$ theorem, is proposed. By analysis, the upper bound on the number of matrix-vector products of block IDR$(s)$ is the same as that of the IDR$(s)$ for a single right-hand side in generic case, i.e., the total number of matrix-vector products of IDR$(s)$ may be $m$ times that of of block IDR$(s)$, where $m$ is the number of right-hand sides. Numerical experiments are presented to show the effectiveness of our proposed method.