Neural machine translation (NMT) has become the dominant approach in the field of machine translation, and is being utilized by most translation services providers. Long Short-Term Memory (LSTM) recurrent neural networks and Transformer neural networks are the two commonly applied NMT modeling mechanisms. Despite their low accuracy, low speed of translation, and involvement of linguistic professionals, other methods of Amharic-Awngi translation such as Statistical Machine Translation (SMT) have already been researched. This work focuses on applying LSTM and Transformer models for Amharic-Awngi translation. Due to an increase in the number of users, the translations of Amharic to Awngi language becomes necessary and to increase the accessibility of the language on the web, it is important to develop machine translation between Amharic and Awngi languages. To address the lack of model training resources, different official documents, news and other written texts are translated. We conduct experiments using two encoder-decoder models. Used a total of 15,111 parallel sentences using both word and morpheme translation units. In order to segment our morpheme data, we have used a Morfessor tool. To propose an optimal model with the best translation unit we consider efficiency (training time, memory usage, and Bilingual Evaluation Understudy (BLEU) score). At time of model training we used BLEU score for evaluation. Among the models explored, the morpheme-based machine translation using the Transformers architecture showed the largest BLEU score at 33.33% on a test set of size 3678. Keywords: Amharic-Awngi Machine translation, LSTM, Transformer, deep learning approach, neural machine translation