Summary: Let \({\mathcal {L}}(X)\) be the Banach algebra of all bounded linear operators on a complex Banach space \(X\). For an operator \(T\in {\mathcal {L}}(X)\), let \(\iota _T(x)\) denote the inner local spectral radius of \(T\) at any vector \(x\) in \(X\). We characterize maps \(\phi \) (not necessarily linear nor surjective) on \({\mathcal {L}}(X)\) which satisfy \[ \begin{aligned} \iota _{T-S} (x)=0 \text{ if and only if }\iota _{\phi (T)-\phi (S)}(x)=0 \end{aligned} \] for every \(x\in X\) and \(T, S\in {\mathcal {L}}(X)\). We also describe surjective linear maps \(\phi \) on \({\mathcal {L}}(X)\) for which \(\phi (I)\) is invertible and either \[ \begin{aligned} \iota _{T}(x)=0 \Longrightarrow \iota _{\phi (T)}(x)=0 \end{aligned} \] for every \(x\in X\) and \(T\in {\mathcal {L}}(X)\), or \[ \begin{aligned} \iota _{\phi (T)}(x)=0 \Longrightarrow \iota _{T}(x)=0 \end{aligned} \] for every \(x\in X\) and \(T\in {\mathcal {L}}(X)\).