High-intensity X-ray free-electron laser (XFEL) beams create transient and non-equilibrium dense states of matter in solid-density targets. These states can be used to develop atomic X-ray lasers with narrow bandwidth and excellent longitudinal coherence, which is not possible with current XFEL pulses. An atomic kinetics model is used to simulate the population dynamics of atomic inner-shell vacancy states in Mg, Al, and Si, revealing the feasibility of population inversion between K-shell and L-shell vacancy states. We also discuss the gain characteristics of these states implying the possibility of atomic X-ray lasers based on inner-shell vacancy states in the 1.5 keV region. The development of atomic X-ray lasers could have applications in high-resolution spectroscopy and nonlinear optics in the X-ray region.