The magnetic ordering and relative structural stability (phase diagram) of La2/3Sr1/3MnO3/SrTiO3 (LSMO/STO) (001) interfaces are considered by performing generalized gradient approximation plus on-site Coulomb correction calculations. A potential antiferromagnetic alignment between the interface layer and bulk is conjectured for the b-type LSMO/STO (001) interface where the atomic-layer stacking near interface is LaOMnO2LaOMnO2SrOMnO2SrOTiO2 (SrO termination). This is in qualitative agreement with the fact that suppression of ferromagnetism at the SrO-terminated (001) interface is found at some experiments. The small magnetic exchange energies, for the SrO-terminated LSMO/STO (001) interfaces, suggest that Mn spins at the interface are likely to reverse as the temperature changes. Hence, possible magnetic reconstructions will be expected to occur at the SrO-terminated (001) interfaces when temperature increases. Under thermodynamic equilibrium, the calculated phase diagrams show that, with the exception of the c-type (001) interface where the atomic-layer stacking near interface is La2/3Sr1/3OMnO2SrOTiO2 (SrO termination), all other considered LSMO/STO (001) interfaces may be stabilized in the proper conditions. The spin-resolved density of states shows that high spin-transport behaviors could be achieved at the ferromagnetic TiO2-terminated (001) interfaces. The results of the calculations can compare favorably with the experimental results for the LSMO/STO (001) junctions.