It is essential to explore the asymptotic stability of the numerical methods for fractional differential equations with delay since their exact solutions are unavailable or difficult to obtain. In this paper, we firstly derive a sufficient condition of the exact solution to be delay-independently asymptotically stable for the space fractional generalized diffusion equation with delay. Next, by applying the linear $$\theta $$-method to temporal dimension, we have established two classes of finite difference schemes independent of the delay for that, which have second-order and fourth-order accuracy in spatial dimension respectively. The unique solvability is proved and the local truncation error is derived. Furthermore, when $$\theta \in [0,1/2)$$, we have obtained the necessary and sufficient condition of the asymptotic stability of both schemes under the specific conditions for the temporal step size $$\Delta t$$ and the spatial step size h. When $$\theta \in [1/2,1]$$, both schemes are unconditionally asymptotically stable. Moreover, the convergence results in the maximum norm are obtained according to the consistency analysis and the Lax theorem. The numerical examples are implemented to verify the theoretical results, which demonstrate that the asymptotic stabilities of both schemes are depending on the parameter $$\theta $$ and the fractional derivative $$\alpha $$, however, independent of the delay $$\tau $$.