•A metric adversarial domain adaptation approach is proposed to successfully achieve cross-domain RUL prediction.•A feature extraction scheme with a dual self-attention module is developed to learn features with multi-scale semantics.•A supervised positive contrastive module is designed to maximize the target-specific mutual information. Many existing domain adaptation-based methods try to derive domain invariant features to address domain shifts and obtain satisfactory remaining useful life (RUL) of bearings under multiple working conditions. However, most methods may not consider local semantics about degradation features and mutual information from target-specific data when aligning distribution discrepancies, thus resulting in limitations. Additionally, the use of contrastive learning to maintain mutual information may introduce unstable negative samples. To overcome these issues, a metric adversarial domain adaptation approach (MADA) is proposed to evaluate the bearing RULs under multiple working conditions. More specifically, an adversarial domain adaptation architecture with a supervised positive contrastive module is developed to consider mutual information without a negative sample, further learning domain invariant features. Also, the dual self-attention module is designed to extract multi-scale contextual semantics between degradation features. Meanwhile, extensive experiments are conducted in twelve cross-domain scenarios for two bearing cases. The experimental results show that the proposed method is more competitive.