Summary: Inexact versions of the block-triangular preconditioners for nonsymmetric matrices of block two-by-two structures presented by \textit{M. F. Murphy, G. H. Golub}, and \textit{A. J. Wathen}[SIAM J. Sci. Comput. 21, No.~6, 1969--1972 (2000; Zbl 0959.65063)] and by \textit{I. C. F. Ipsen} [ibid. 23, No.~3, 1050--1051 (2001; Zbl 0998.65049)] are presented, and the two preconditioners for symmetric block two-by-two matrices presented by \textit{C. Durazzi} and \textit{V. Ruggiero} [ Numer. Linear Algebra Appl. 10, No.~8, 673--688 (2003)] are extended to general nonsymmetric matrices. Moreover, we precisely describe the spectral properties of the preconditioned matrices and the finite-step termination properties of the preconditioned Krylov subspace iteration methods with an optimal or Galerkin property, with respect to these preconditioners. Several numerical examples are performed to illustrate the effectiveness of the proposed preconditioners.