A quadratic spline collocation method combined with the Crank-Nicolson time discretization of the space fractional diffusion equations gives discrete linear systems, whose coefficient matrix is the sum of a tridiagonal matrix and two diagonal-multiply-Toeplitz-like matrices. By exploiting the Toeplitz-like structure, we split the Toeplitz-like matrix as the sum of a Toeplitz matrix and a rank-2 matrix and Strang’s circulant preconditioner is constructed to accelerate the convergence of Krylov subspace method like generalized minimal residual method for solving the discrete linear systems. In theory, both the invertibility of the proposed preconditioner and the clustering spectrum of the corresponding preconditioned matrix are discussed in detail. Finally, numerical results are given to demonstrate that the performance of the proposed preconditioner is better than that of the generalized T. Chan’s circulant preconditioner proposed recently by Liu et al. (J. Comput. Appl. Math., 360 (2019), pp. 138–156) for solving the discrete linear systems of one-dimensional and two-dimensional space fractional diffusion equations.