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Recently, deep learning (DL)-based non-intrusive speech assessment models have attracted great attention. Many studies report that these DL-based models yield satisfactory assessment performance and good flexibility, but their performance in unseen environments remains a challenge. Furthermore, compared to quality scores, fewer studies elaborate deep learning models to estimate intelligibility scores. This study proposes a multi-task speech intelligibility prediction model, called MTI-Net, for simultaneously predicting human and machine intelligibility measures. Specifically, given a speech utterance, MTI-Net is designed to predict human subjective listening test results and word error rate (WER) scores. We also investigate several methods that can improve the prediction performance of MTI-Net. First, we compare different features (including low-level features and embeddings from self-supervised learning (SSL) models) and prediction targets of MTI-Net. Second, we explore the effect of transfer learning and multi-tasking learning on training MTI-Net. Finally, we examine the potential advantages of fine-tuning SSL embeddings. Experimental results demonstrate the effectiveness of using cross-domain features, multi-task learning, and fine-tuning SSL embeddings. Furthermore, it is confirmed that the intelligibility and WER scores predicted by MTI-Net are highly correlated with the ground-truth scores.