Large scale applications running on HPC systems often require a substantial amount of memory and can have a large computational overhead. Lossy data compression techniques can reduce the size of the data and associated communication cost, but the effect of the loss ofaccuracy on the numerical algorithm can be hard to predict. In this paper we examine the FGMRES algorithm, which requires the storage of a basis for the Krylov subspace and for the search space spanned by the solutions of the preconditioning systems. We show that the vectors spanning this search space can be compressed by looking at the combination of FGMRES and compression in the context of inexact Krylov subspace methods. This allows us to derive a bound on the normwise relative compression error in each iteration. We use this bound to formulate a number of different practical compression strategies, and validate and compare them through numerical experiments.