Given feasible strongly regular graph parameters $(v,k,λ,μ)$ and a non-negative integer $d$, we determine upper and lower bounds on the order of a $d$-regular induced subgraph of any strongly regular graph with parameters $(v,k,λ,μ)$. Our new bounds are at least as good as the bounds on the order of a $d$-regular induced subgraph of a $k$-regular graph determined by Haemers. Further, we prove that for each non-negative integer $d$, our new upper bound improves on Haemers' upper bound for infinitely many strongly regular graphs.