This work proposes a method to asses the local asymptotic stability and to provide polyhedral estimates of the region of attraction of the origin (RAO) of linear systems under aperiodic sampled-data control and saturating inputs. The approach is based on a discrete-time model that describes the behavior of the system state between consecutive sampling instants. It corresponds to a difference inclusion defined from a partition of the intersampling interval and from the saturated and nonsaturated (SNS) embedding of saturation functions. A method to construct a contractive polyhedral set for this model is proposed. It is shown that this set induces a local Lyapunov function strictly decreasing at the sampling instants and that it is an estimate of the RAO of the continuous-time closed-loop system.