The Tolman VII solution is considered by some as one of the few analytical solutions to Einstein's equations, which describes approximately well the interior of neutron stars (NSs). This solution is characterized by the mass $M$, radius $R$, and an energy density that varies quadratically with the radial coordinate $r$. Recently, Jiang and Yagi proposed a modification of this solution, the so-called modified Tolman VII (MTVII) solution, by introducing an additional quartic term to the energy density radial profile. The MTVII solution is an approximate solution to Einstein's equation, which includes a new parameter $α$ that allows the solution to have a better agreement with the energy density profiles for realistic NSs. Here we consider the MTVII solution, showing that for certain values of the parameter $α$ and compactness $\mathcal{C}$ this solution manifests a region of negative pressure near the surface which leads to negative values of the tidal Love number. To alleviate these drawbacks, we introduce an exact version of the MTVII solution obtained by solving numerically Einstein's equations for the MTVII energy density profile. As an application of our new exact MTVII (EMTVII) solution, we calculate the tidal Love number and tidal deformability, as a function of $\mathcal{C}$, for different values of the parameter $α$. We find that the EMTVII solution predicts a positive tidal Love number for the whole range of allowed values of parameters $(\mathcal{C},α)$, in agreement with previous results for realistic NSs.

10 pages, 10 figures. Matches version accepted for publication in Phys. Rev. D