Speech emotion recognition is a challenging but important task in human computer interaction (HCI). As technology and understanding of emotion are progressing, it is necessary to design robust and reliable emotion recognition systems that are suitable for real-world applications both to enhance analytical abilities supporting human decision making and to design human-machine interfaces (HMI) that assist efficient communication. This paper presents a multimodal approach for speech emotion recognition based on Multi-Level Multi-Head Fusion Attention mechanism and recurrent neural network (RNN). The proposed structure has inputs of two modalities: audio and text. For audio features, we determine the mel-frequency cepstrum (MFCC) from raw signals using the OpenSMILE toolbox. Further, we use pre-trained model of bidirectional encoder representations from transformers (BERT) for embedding text information. These features are fed parallelly into the self-attention mechanism base RNNs to exploit the context for each timestamp, then we fuse all representatives using multi-head attention technique to predict emotional states. Our experimental results on the three databases: Interactive Emotional Motion Capture (IEMOCAP), Multimodal EmotionLines Dataset (MELD), and CMU Multimodal Opinion Sentiment and Emotion Intensity (CMU-MOSEI), reveal that the combination of the two modalities achieves better performance than using single models. Quantitative and qualitative evaluations on all introduced datasets demonstrate that the proposed algorithm performs favorably against state-of-the-art methods. INDEX TERMS Speech emotion recognition, multi-level multi-head fusion attention, RNN, audio features, textual features.