We investigate the existence and propagation of solitons in a long-range extension of the quartic Fermi–Pasta–Ulam (FPU) chain of anharmonic oscillators. The coupling in the linear term decays as a power-law with an exponent $1<\alpha \leqslant 3$ . We obtain an analytic perturbative expression of traveling envelope solitons by introducing a non linear Schrödinger equation for the slowly varying amplitude of short wavelength modes. Due to the non analytic properties of the dispersion relation, it is crucial to develop the theory using discrete difference operators. Those properties are also the ultimate reason why kink-solitons may exist but are unstable, at variance with the short-range FPU model. We successfully compare these approximate analytic results with numerical simulations for the value $\alpha =2$ which was chosen as a case study. © 2017 IOP Publishing Ltd Printed in the UK.