According to a letter sent to The Independent, Dr Pachauri, chairman of the Intergovernmental Panel on Climate Change (IPCC), and McCartney, who are both vegetarians, blame the worsening global warming on a rise in the number of people who eat meat.
They believe that global food shortages are exacerbated by the planting of cereal crops for animal fodder. A mass switch to a more vegetarian diet will, according to them, help the poorest people in the world.
Becoming vegetarian, or at the very least eating less red meat, is the single most effective act anyone can take to lessen greenhouse gas emissions, said Dr Pachauri and McCartney.
As well as producing the greenhouse gas methane, the livestock business uses up increasingly scarce sources of fresh water and increases other forms of pollution through its need for agricultural chemicals, they argue.
Dr Pachauri and McCartney cite a 2006 report by the UN's Food and Agriculture Organization, which stated that livestock are one of the most significant contributors to climate change because 70 per cent of former forests in the Amazon have been turned over to grazing and livestock now use 30 per cent of the world's land surface.
Unfortunately, with higher incomes, societies, even in developing countries, are turning to greater consumption of animal protein, which reduces the availability of food grains for direct consumption by impoverished human beings, they said.
Already 60 per cent of food crop production in North America and western Europe is being diverted for production of meat, they added.
The letter also quoted them as saying, Citizens across the world often ask what it is that they can do to mitigate emissions. There are several reasons for a shift to a much lower input of meat in human diets if not complete vegetarianism.
We are writing this letter not because vegetarianism is a fad or an emotional issue, but because it is a very attractive option for reducing emissions of greenhouse gases and stabilising the Earth's climate and ensuring global food security, it added.
Dr Pachauri, who accepted a half-share in this year's Nobel Peace Prize on behalf of the IPCC, has long advocated vegetarianism as a way of fighting climate change.
He has been a vegetarian for eight years, while Sir Paul stopped eating meat about 30 years ago, largely because of his concerns about the welfare of farm animals.