The penetrable-square-well (PSW) pair interaction potential is defined as $\phi (r)=\epsilon_r$ if the two interacting particles are overlapped ($r<\sigma$), $\phi(r)=-\epsilon_a$ inside a corona ($\sigma \lambda$). Thus, the potential reduces to the conventional square-well (SW) one in the limit $\epsilon_r\to\infty$ and to the penetrable-sphere (PS) potential if $\epsilon_a\to0$ or $\lambda\to\sigma$. This paper aims at studying the temperature dependence of the Navier--Stokes transport coefficients of a dilute gas of particles interacting via the PSW model. By exploiting the fact that the PSW scattering process is analogous to that of a light ray passing through two concentric spherical media with different refractive indices, the scattering angle is analytically derived as a function of the impact parameter and the relative velocity of the colliding particles; depending on the values of those two quantities, collisions can be soft, hard, or grazing. Next, by standard application of known general results from the Chapman--Enskog method, the Navier--Stokes transport coefficients in the first-order approximation are numerically evaluated. It is found that the PSW coefficients are practically indistinguishable from the SW ones for temperatures low enough ($k_BT\lesssim 0.2 \epsilon_r$), there exists a transition regime ($0.2 \epsilon_r\lesssim k_BT\lesssim 10\epsilon_r$) where the transport coefficients interpolate between the SW and the PS ones, and finally the PSW coefficients are comparable to the PS ones for high enough temperatures ($k_BT\gtrsim 10\epsilon_r$). The results are applied to the temperature profiles of the planar Fourier flow., Comment: 10 pages, 6 figures; contributed paper at the 31st International Symposium on Rarefied Gas Dynamics (Glasgow, UK, July 23-27, 2018); v2: new section added