Complete active space-multiconfiguration self-consistent-field (CAS-MCSCF) followed by configuration interaction calculations, which included up to 250 000 configurations, are carried out on 16 electronic states of Ga4 and In4. Three nearly-degenerate electronic states of 3B3u, 3A(u), and 1A(g) symmetries with rhombus, square, and square geometries, respectively, are found as candidates for the ground states of Ga4 and In4. The tetramers of Ga and In are bound by 1.2 and 1 eV, respectively, compared to the trimers.