Abstract : In matrix analysis, the \textit{Wielandt-Mirsky conjecture} states that $$ \dist(\sigma(A), \sigma(B)) \leq \|A-B\| $$ for any normal matrices $ A, B \in \mathbb C^{n\times n}$ and any operator norm $\|\cdot \|$ on $C^{n\times n}$. Here $\dist(\sigma(A), \sigma(B))$ denotes the optimal matching distance between the spectra of the matrices $A$ and $ B$. It was proved by A.~J.~Holbrook (1992) that this conjecture is \textit{false } in general. However it is true for the Frobenius distance and the Frobenius norm (the \textit{Hoffman-Wielandt inequality}). The main aim of this paper is to study the Hoffman-Wielandt inequality and some weaker versions of the Wielandt-Mirsky conjecture for matrix polynomials.