Facing asymmetric carbon abatement information of the energy service company (ESCO), the manufacturer not only suffers from economic losses, but also is unable to achieve the optimum emission reduction effect. For a manufacturer, designing incentive contracts plays an essential role in reducing losses resulted from ESCO information asymmetry. However, previous studies have ignored the manufacturer's contract design issue when facing different types of ESCO with high or low abatement efficiency, and disregarded manufacturer with different goals in choosing a contract. To fill this research gap, this study aims to investigate the manufacturer's contract design issue and explore the issue of the manufacturer's contract selection according to its different goals. To achieve this objective, this research focuses on the embedded low-carbon service supply chain, including a manufacturer and an ESCO with different carbon abatement efficiency. First, this study develops the full information model (the basic model). Furthermore, this research considers the information asymmetry of different ESCO's carbon abatement efficiency to develop the screening and pooling contract models. This study comparatively analyzes the contract selection of the manufacturer with different goals, including carbon reduction level, profit, or cooperation period. The results show that when cooperating with high-efficiency ESCO, the manufacturer providing the screening contract could obtain a higher carbon reduction level compared with providing the pooling contract. In contrast, when cooperating with low-efficiency ESCO, the manufacturer providing the pooling contract would obtain a higher carbon reduction level. The manufacturer should choose the pooling contract if pursuing higher profit. In addition, the manufacturer should choose to provide the pooling contract if seeking long-term cooperation. The findings are helpful for the manufacturer's decision-making to cooperate with ESCO under the asymmetric information of carbon abatement.