Based on the relaxed alternating positive semi-definite splitting (RAPSS) preconditioner, in this paper, a new preconditioner, called variant of relaxed alternating positive semi-definite splitting (VRAPSS) preconditioner, is presented and discussed. Spectral properties of the VRAPSS preconditioned matrix are analyzed in detail. Numerical experiments are provided to verify the efficiency of the VRAPSS preconditioner. © 2021, The JJIAM Publishing Committee and Springer Japan KK, part of Springer Nature.