In this work, deep learning (DL)-based transmit antenna selection (TAS) strategies are employed to enhance the average bit error rate (ABER) and energy efficiency (EE) performance of a spectrally efficient fully generalized spatial modulation (FGSM) scheme. The Euclidean distance-based antenna selection (EDAS), a frequently employed TAS technique, has a high search complexity but offers optimal ABER performance. To address TAS with minimal complexity, we present DL-based approaches that reframe the traditional TAS problem as a classification learning problem. To reduce the energy consumption and latency of the system, we presented three DL architectures in this study, namely a feed-forward neural network (FNN), a recurrent neural network (RNN), and a 1D convolutional neural network (CNN). The proposed system can efficiently process and make predictions based on the new data with minimal latency, as DL-based modeling is a one-time procedure. In addition, the performance of the proposed DL strategies is compared to two other popular machine learning methods: support vector machine (SVM) and K-nearest neighbor (KNN). While comparing DL architectures with SVM on the same dataset, it is seen that the proposed FNN architecture offers a ~3.15% accuracy boost. The proposed FNN architecture achieves an improved signal-to-noise ratio (SNR) gain of ~2.2 dB over FGSM without TAS (FGSM-WTAS). All proposed DL techniques outperform FGSM-WTAS.