Let ${gamma}$ be a hyperbolic closed orbit of a $C^1$ vector field X on a compact boundaryless Riemannian manifold M, and let $C_X({gamma})$ be the chain component of X which contains ${gamma}$. We say that $C_X({gamma})$ is $C^1$ robustly shadowable if there is a $C^1$ neighborhood $mathcal{U}$ of X such that for any $Y{in}mathcal{U}$, $C_Y({gamma}_Y)$ is shadowable for $Y_t$, where ${gamma}_Y$ denotes the continuation of ${gamma}$ with respect to Y. In this paper, we prove that any $C^1$ robustly shadowable chain component $C_X({gamma})$ does not contain a hyperbolic singularity, and it is hyperbolic if $C_X({gamma})$ has no non-hyperbolic singularity.