Real-world machine-learning applications require robust models that generalize well to distribution shift settings, which is typical in real-world situations. Domain adaptation techniques aim to address this issue of distribution shift by minimizing the disparities between domains to ensure that the model trained on the source domain performs well on the target domain. Nevertheless, the existing domain adaptation methods are computationally very expensive. In this work, we aim to improve the efficiency of existing supervised domain adaptation (SDA) methods by using a subset of source data that is similar to target data for faster model training. Specifically, we propose ORIENT, a subset selection framework that uses the submodular mutual information (SMI) functions to select a source data subset similar to the target data for faster training. Additionally, we demonstrate how existing robust subset selection strategies, such as GLISTER, GRADMATCH, and CRAIG, when used with a held-out query set, fit within our proposed framework and demonstrate the connections with them. Finally, we empirically demonstrate that SDA approaches like d-SNE, CCSA, and standard Cross-entropy training, when employed together with ORIENT, achieve a) faster training and b) better performance on the target data.