The ground states of the nuclei ${}^{40}\mathrm{Mg}$ and ${}^{39}\mathrm{Na}$ are investigated using the hyperspherical formalism. Since they are located at the edge of the “big island of inversion”, we concentrate on whether we are likely to find two-neutron Borromean halos in these nuclei. A three-body model with effective n-n and ${}^{38}\mathrm{Mg}+n$ interactions is built for ${}^{40}\mathrm{Mg}$ based on the available data. We also give predictions for the low-lying spectrum of ${}^{38}\mathrm{Na}{=}^{37}\mathrm{Na}+n$ and two-neutron separation energy of the ${}^{39}\mathrm{Na}$ nucleus. Depending on parameter choice, we report an increase in the matter radii in the range 0.1-0.5 fm relative to those of the core nuclei. The results suggest a two-neutron halo structure in ${}^{40}\mathrm{Mg}$ for a subset of parameters, reinforcing the prediction of a Borromean halo nucleus. The calculations indicate that a two-neutron halo is even more likely for ${}^{39}\mathrm{Na}$. As expected, the halo is linked to the disappearance of the shell gap in these nuclei due to the inversion of the $2p_{3/2}$ and $1f_{7/2}$ orbitals. We study the total cross section for scattering of these nuclei from a carbon target using a Glauber model and show that these provide a clear signal to assess the halo structure.