Suprathermal electrons are routinely generated in high-intensity laser produced plasmas via instabilities driven by non-linear laser-plasma interaction. Their accurate characterization is crucial for the performance of inertial confinement fusion as well as for performing experiments in laboratory astrophysics and in general high-energy-density physics. Here, we present studies of non-thermal atomic states excited by suprathermal electrons in kJ-ns-laser produced plasmas. Highly spatially and spectrally resolved X-ray emission from the laser-deflected part of the warm dense Cu foil visualized the hot electrons. A multi-scale two-dimensional hydrodynamic simulation including non-linear laser-plasma interactions and hot electron propagation has provided an input for ab initio non-thermal atomic simulations. The analysis revealed a significant delay between the maximum of laser pulse and presence of suprathermal electrons. Agreement between spectroscopic signatures and simulations demonstrates that combination of advanced high-resolution X-ray spectroscopy and non-thermal atomic physics offers a promising method to characterize suprathermal electrons inside the solid density matter.