We describe a method for locating and following saddle-node bifurcations of invariant circles in quasiperiodically forced systems. This is based on making successive rational approximations to the quasiperiodic forcing, and computing the locations of saddle-node bifurcations of appropriate periodic orbits for the approximating system. An example of the application of the algorithm to the quasiperiodically forced sine map is given. The method is also equally applicable to period doubling bifurcations of invariant circles.