For nonsingular indefinite matrices of saddle-point (or KKT) form, Murphy, Golub and Wathen (2000) have proposed an « ideal » block diagonal preconditioner based on the exact Schur complement. In this talk, assuming a zero (2,2) block, we focus on the case where the (1,1) block is symmetric positive definite and (eventually) very badly conditioned, but with only a few very small eigenvalues. Under the assumption that a good approximation of these eigenvalues and their associated eigenvectors is available, we consider different approximations of the block diagonal preconditioner of Murphy, Golub and Wathen. We analyze the spectral properties of the preconditioned matrices and show how it is possible to appropriately recombine the available spectral information from the (1,1) block through a particular Schur complement approximation that allows to build an efficient block diagonal preconditioner with little extra cost. We finally illustrate the performance of the proposed preconditioners with some numerical experiments.