This paper focuses on a green supply chain consisting of a dominant retailer and a capital-constrained supplier. Two different shortage situations of the supplier’s internal funds are considered, namely, only lack of green investment funds and lack of both green investment and production funds. The supplier can use green order financing to solve its financial problems, that is, to apply for financing from the bank in virtue of the green order signed with the downstream retailer. Besides, in order to promote the supplier’s green production and reduce its operating costs, the government provides two types of green subsidy strategies, i.e., green investment subsidy (GIS) for the supplier and green credit subsidy (GCS) for the bank. In this study, we mainly investigate the optimal subsidy strategies of the government under the objective of “equal-product-greenness”. Results show that under different capital-constrained degrees of the supplier, the introduction of GIS or GCS can improve the product greenness, the product sales, the supply chain members’ profits, the consumer surplus as well as the environmental benefits. No matter which subsidy strategy is adopted, when the supplier lacks both of the green investment and production funds, all the performance of the above indicators will be reduced. Besides, under the objective of “equal-product-greenness”, GIS and GCS can achieve the same social welfare, while GIS would be more efficient. Finally, taking the case that the supplier lacks only the green investment funds as an example, we further extend the problem to explore the optimal choices of the government’s subsidy strategies under the new objective of “equal subsidy amount”, and either GIS or GCS may be more efficient in certain circumstances.