In the present paper, the theory of multi-phase-lags with fractional derivative heat transfer thermoelasticity was used to study the wave propagation on a nonlocal fiber-reinforced thermoelastic medium. The exact expressions for the displacement components, temperature, and stress components are obtained by using the Fourier and Laplace transforms. The comparisons are shown graphically to explore the effects of rotation, initial stress and fractional order parameters as well as a nonlocal parameter on different physical quantities. Numerical computations are carried out with the help of Matlab software.