For a block three-by-three nonsymmetric positive semi-definite matrix that appears in many applications or in the study of preconditioning techniques, we estimate the lower and upper bounds of the real and the imaginary parts of eigenvalues of this matrix. The bounds are expressed in terms of the extreme eigenvalues or singular values of the associated block matrices. Numerical experiments demonstrate that our estimated bounds are relatively sharp and can indeed predict the effectiveness of the preconditioner.