This paper deals with a generalization of Geršgorin’s theorem. It is well known that the spectrum of a matrix \(A\) belongs to \(\Gamma(A)\cap\Gamma(A^{T})\) provided that \(\Gamma(A)\) and \(\Gamma(A^{T})\) are the Geršgorin sets of \(A\) and \(A^{T}\), respectively. Using classes of nonsigular H-matrices the authors obtain conditions on the spectrum to lie in an extension of \(\bigcup_i \Gamma_i(A)\cap\Gamma_i(A^{T})\), provided that \(\Gamma_i(.)\) are the respective Geršgorin disks, so that this new area is included in the classical one. Several examples illustrate this interesting new technique.