In today's worldwide markets, supply chain members have to control carbon emissions for a clean environment. This study addresses a dual-channel green supply chain model by contemplating dual selling channels, carbon diminution rate and online delivery lead-time as marketing efforts to fascinate customers to buy more products. Moreover, the carbon tax protocol, green technology, carbon cap-and-trade protocol are applied to the supply chain to save the environment by reducing carbon emissions. The customers' demands for both the direct and traditional retail channels are dependent on direct selling price, retail price, carbon reduction rate, and delivery lead-time of online products. The objective of this work is to maximise the supply chain profit by minimising the amount of carbon emissions. To determine optimal outcomes, a centralised system, manufacturer-leadership Stackelberg game and retailer-leadership Stackelberg game are employed. A comparison among the centralised system and the two Stackelberg games is also carried out. The results presented that to enhance the profits of the manufacturer as well as retailer and to decline carbon emissions, manufacturer-leadership Stackelberg game is the adequate game policy. Some managerial inspirations of this proposed work are listed. The paper ends with conclusions and possible future research directions.