Suppose that a linear bounded operator B on a Hilbert space exhibits at least linear GMRES convergence, i.e., there exists Mp < 1 such that the GMRES residuals fulfill ||rfc|| < MB||rk-1|| f°r every initial residual ro and step k E N. We prove that GMRES with a compactly perturbed operator A = B + C admits the bound ||rkll < rij=i (Mb + (1 + Mb) IM_1II (C*)), i.e., the singular values αj(C) control the departure from the bound for the unperturbed problem. This result can be seen as an extension of [I. Moret, SIAM J. Numer. Anal., 34 (1997), pp. 513-516], where only the case B = XI is considered. In this special case Mp = 0 convergence is superlinear. [ABSTRACT FROM AUTHOR]