We analyze the $f_0(500)$ state generated as a pole of $\pi\pi$ scattering within unitarized low-energy effective theories at finite temperature. The relation of that thermal pole with the scalar susceptibility is studied within a scalar saturation approach, which yields results complying with lattice data. The robustness and predictability of this method are studied in terms of the low-energy constants involved and the unitarization method. A detailed fit to lattice data is provided, which is compared to a Hadron Resonance Gas description. Our analysis highlights the importance of this thermal state to describe the main qualitative features of the scalar susceptibility around the chiral transition.