Abstract An inexact variant of inverse subspace iteration is used to find a small invariant pair of a large quadratic matrix polynomial. It is shown that linear convergence is preserved provided the inner iteration is performed with increasing accuracy. A preconditioned block GMRES solver is employed as inner iteration. The preconditioner uses the strategy of “tuning” which prevents the inner iteration from increasing and therefore results in a substantial saving in costs. The accuracy of the computed invariant pair can be improved by the addition of a post-processing step involving very few iterations of Newton’s method. The effectiveness of the proposed approach is demonstrated by numerical experiments.