In this paper, we analyze the modified $$f({\mathcal {R}})$$ $f\left(R\right)$ gravity models in Friedmann–Lemaître–Robertson–Walker (FLRW) background. The actions of bouncing cosmology are studied under consideration of different viable models in $$f({\mathcal {R}})$$ $f\left(R\right)$ gravity theory that can resolve the difficulty of singularity in standard Big-Bang cosmology. Under different viable models in $$f({\mathcal {R}})$$ $f\left(R\right)$ gravity theory, the cosmological constraints are plotted in provisions of cosmic-time, then investigated the bounce circumstance. In addition, the red-shift parameter is used to reconstruct the modified gravity, and compile the cosmological parameters that infer accelerated universe expansion. Finally, the situation stability is evaluated with a sound speed feature, which illustrates late-time stability.