The observational proxy estimates suggest that the North Atlantic overturning stream function associated with the North Atlantic Deep Water (NADW) production and outflow was substantially weaker during the last glacial maximum (LGM) than that observed under present conditions. The impact of the changes in overturning circulation on the glacial carbon budget is investigated using a box model. The carbon box model reveals that the atmospheric $CO_2$ concentration is more sensitive to change in the overturning circulation of the North Atlantic than that of the Southern Ocean, especially when North Atlantic overturning becomes weaker. For example, when the strength of the North Atlantic overturning circulation is halved, the atmospheric $CO_2$ concentration is reduced by 50ppm of that associated with the accumulation of $CO_2$ in the deep ocean. This result implies that a weaker North Atlantic overturning circulation may play an important role in the lowering of LGM atmospheric $CO_2$ concentration.