The paper presents a forecast of changes in the methane content in the air in surface layer of the atmosphere. The forecast was made by the models based on the two most common types of artificial neural networks (ANN): A nonlinear autoregressive neural networks with exogenous inputs (NARX) and Elman neural network (ENN). For training, we used the Levenberg–Marquardt learning algorithm. The data were collected upon monitoring the greenhouse gases on Bely Island, Yamal-Nenets Autonomous Okrug, Russia. For the comparison, the three time intervals with the different patterns of changes in methane content were chosen. To assess the prediction accuracy of the models, we used the mean absolute error, mean square error, and the standardized measure of the model prediction error degree—the index of agreement. The model based on the artificial neural network NARX for all simulated intervals was the most accurate.