In this paper, we address the problem of predicting a response variable in the context of both, spatially correlated and high-dimensional data. To reduce the dimensionality of the predictor variables, we apply the sufficient dimension reduction (SDR) paradigm, which reduces the predictor space while retaining relevant information about the response. To achieve this, we impose two different spatial models on the inverse regression: the separable spatial covariance model (SSCM) and the spatial autoregressive error model (SEM). For these models, we derive maximum likelihood estimators for the reduction and use them to predict the response via nonparametric rules for forward regression. Through simulations and real data applications, we demonstrate the effectiveness of our approach for spatial data prediction.