We determine the constraints imposed on the 10d target superspace geometry by the requirement of classical kappa-symmetry of the Green-Schwarz superstring. In the type I case we find that the background must satisfy a generalization of type I supergravity equations. These equations depend on an arbitrary vector X a and imply the one-loop scale invariance of the GS sigma model. In the special case when X a is the gradient of a scalar ϕ (dilaton) one recovers the standard type I equations equivalent to the 2d Weyl invariance conditions of the superstring sigma model. In the type II case we find a generalized version of the 10d supergravity equations the bosonic part of which was introduced in arXiv:1511.05795. These equations depend on two vectors X a and K a subject to 1st order differential relations (with the equations in the NS-NS sector depending only on the combination X a = X a + K a ). In the special case of K a = 0 one finds that X a = ∂ a ϕ and thus obtains the standard type II supergravity equations. New generalized solutions are found if K a is chosen to be a Killing vector (and thus they exist only if the metric admits an isometry). Non-trivial solutions of the generalized equations describe K-isometric backgrounds that can be mapped by T-duality to type II supergravity solutions with dilaton containing a linear isometry-breaking term. Examples of such backgrounds appeared recently in the context of integrable η-deformations of AdS n × S n sigma models. The classical kappa-symmetry thus does not, in general, imply the 2d Weyl invariance conditions for the GS sigma model (equivalent to type II supergravity equations) but only weaker scale invariance type conditions.