Governments commonly utilize subsidy policy to incentivize manufacturers to produce green products, promoting sustainable development. However, in the presence of information asymmetry, some manufacturers may dishonestly misrepresent the green degree of their products to secure higher subsidies. This study examines different incentive contracts between the government and a green product manufacturer who keeps private information of a product's green-degree in a principal-agent model. Lump-sum transfer and fixed- and flexible-proportion benefit-sharing contracts are proposed to investigate screening and improving green-degree issues. To further enhance the flexible-proportion benefit-sharing contract, we construct a non-linear coordinated contract based on the Nash bargaining solution. The revelation principle and Nash bargaining are performed for comparison and analysis of the contracts. We find that the lump-sum contract reveals true green-degree information but fails to impel manufacturers to improve product's green-degree in developing countries where green product development is in initial stages. In contrast, both fixed- and flexible- proportion benefit-sharing contracts are effective in reveling and enhancing green-degree. The non-linear coordination contract optimizes resource allocation and achieves Pareto improvement. An applied case study for inkjet printer operations and numerical experiments corroborate our model findings.