The Department of Energy and Climate Change is spending £6m on the ad, which was created by AMV BBDO, to promote its Act on CO2 carbon reduction initiative.
The ad uses the device of a children's storybook, which shows a British town deep under water, with people and animals drowning. Carbon dioxide is shown rising from cars, homes and everyday appliances in clouds of black soot, which then form a jagged-toothed monster.
According to an ASA spokesman, the complaints include objections about the existence of climate change, objections to the ad's claim that climate change is man-made, and questions as to whether there is a scientific consensus about it.
In relation to the second point, the ad includes the line: "The grown-ups discovered that over 40% of the CO2 comes from everyday things like keeping houses warm and driving cars".
While most complaints claim the ad is misleading, some have also said it is frightening for children.
The ASA is expecting to make a decision in the next week as to whether there are grounds for investigating the ad.
The original Brand Republic story about the ad campaign was the most commented article of the last week.