Subspace methods for system identification estimate the dynamics of state-space models either by using the ’shift-invariance’ property of an estimated observability or controllability matrix, or by estimating a state sequence and then solving a least-squares problem to obtain the system matrices. In either case it is possible for the estimated system to be unstable. We present algorithms to find stable approximants to a least-squares problem, which can then be applied to subspace methods to ensure stability. Either asymptotic or marginal stability can be ensured, in the latter case a pole or a pair of poles being forced to lie on the unit circle. In addition, some results on a sufficient condition for stability for least-squares solutions obtained by the shift invariance approach are derived.