We present DFO-GN, a derivative-free version of the Gauss–Newton method for solving nonlinear least-squares problems. DFO-GN uses linear interpolation of residual values to build a quadratic model of the objective, which is then used within a typical derivative-free trust-region framework. We show that DFO-GN is globally convergent and requires at most $O(\epsilon^{-2})$ iterations to reach approximate first-order criticality within tolerance $\epsilon$. We provide an implementation of DFO-GN and compare it to other state-of-the-art derivative-free solvers that use quadratic interpolation models. We demonstrate numerically that despite using only linear residual models, DFO-GN performs comparably to these methods in terms of objective evaluations. Furthermore, as a result of the simplified interpolation procedure, DFO-GN has superior runtime and scalability. Our implementation of DFO-GN is available at https://github.com/numericalalgorithmsgroup/dfogn (https://doi.org/10.5281/zenodo.2629875).