We introduce a selection model-based imputation approach to be used within the Fully Conditional Specification (FCS) framework for the Multiple Imputation (MI) of incomplete ordinal variables that are supposed to be Missing Not at Random (MNAR). Thereby, we generalise previous work on this topic which involved binary single-level and multilevel data to ordinal variables. We apply an ordered probit model with sample selection as base of our imputation algorithm. The applied model involves two equations that are modelled jointly where the first one describes the missing-data mechanism and the second one specifies the variable to be imputed. In addition, we develop a version for hierarchical data by incorporating random intercept terms in both equations. To fit this multilevel imputation model we use quadrature techniques. Two simulation studies validate the overall good performance of our single-level and multilevel imputation methods. In addition, we show its applicability to empirical data by applying it to a common research topic in educational science using data of the National Educational Panel Study (NEPS) and conducting a short sensitivity analysis. Our approach is designed to be used within the R software package mice which makes it easy to access and apply.