Introduction: Electroencephalogram (EEG) signals have gained significant popularity in various applications due to their rich information content. However, these signals are prone to contamination from various sources of artifacts, notably the electrooculogram (EOG) artifacts caused by eye movements. The most effective approach to mitigate EOG artifacts involves recording EOG signals simultaneously with EEG and employing blind source separation techniques, such as independent component analysis (ICA). Nevertheless, the availability of EOG recordings is not always feasible, particularly in pre-recorded datasets. Objective: In this paper, we present a novel methodology that combines a long short-term memory (LSTM)-based neural network with ICA to address the challenge of EOG artifact removal from contaminated EEG signals. Approach: Our approach aims to accomplish two primary objectives: 1) estimate the horizontal and vertical EOG signals from the contaminated EEG data, and 2) employ ICA to eliminate the estimated EOG signals from the EEG, thereby producing an artifact-free EEG signal. Main results: To evaluate the performance of our proposed method, we conducted experiments on a publicly available dataset comprising recordings from 27 participants. We employed well-established metrics such as mean squared error, mean absolute error, and mean error to assess the quality of our artifact removal technique. Significance: Furthermore, we compared the performance of our approach with two state-of-the-art deep learning-based methods reported in the literature, demonstrating the superior performance of our proposed methodology.