This study proposes a robust outlier detection method based on the circular median for non-parametric linear-circular regression in case the response variable includes outlier(s) and the residuals are Wrapped-Cauchy distributed. Nadaraya-Watson and local linear regression methods were employed to obtain non-parametric regression fits. The proposed method’s performance was investigated by using a real dataset and a comprehensive simulation study with different sample sizes, contamination, and heterogeneity degrees. The method performs quite well in medium and higher contamination degrees, and its performance increases as the sample size and the homogeneity of data increase. In addition, when the response variable of linear-circular regression contains outliers, the Local Linear Estimation method fits the data set better than the Nadaraya Watson method.