Salinity is a leading threat to crop growth throughout the world. Salt stress induces altered physiological processes and several inhibitory effects on the growth of cereals, including wheat (Triticum aestivum L.). In this study, we determined the effects of salinity on five spring and five winter wheat genotypes seedlings. We evaluated the salt stress on root and shoot growth attributes, i.e., root length (RL), shoot length (SL), the relative growth rate of root length (RGR-RL), and shoot length (RGR-SL). The ionic content of the leaves was also measured. Physiological traits were also assessed, including stomatal conductance (gs), chlorophyll content index (CCI), and light-adapted leaf chlorophyll fluorescence, i.e., the quantum yield of photosystem II (Fv′/Fm′) and instantaneous chlorophyll fluorescence (Ft). Physiological and growth performance under salt stress (0, 100, and 200 mol/L) were explored at the seedling stage. The analysis showed that spring wheat accumulated low Na+ and high K+ in leaf blades compared with winter wheat. Among the genotypes, Sakha 8, S-24, W4909, and W4910 performed better and had improved physiological attributes (gs, Fv′/Fm′, and Ft) and seedling growth traits (RL, SL, RGR-SL, and RGR-RL), which were strongly linked with proper Na+ and K+ discrimination in leaves and the CCI in leaves. The identified genotypes could represent valuable resources for genetic improvement programs to provide a greater understanding of plant tolerance to salt stress.