The government has said the species, which has only become widespread in the UK since its introduction from America in the late 19th century, costs the economy £1.8bn a year.
The plan to reduce their numbers involves lacing hazelnut spread with contraceptives, and placing the food in feeding pots only the squirrels can access.
Environment minister Zac Goldsmith has said the rising numbers of grey squirrels, which are approaching 3 million, threaten efforts to tackle the climate crisis, as they damage trees through bark stripping.
Their arrival in the UK has also had a “disastrous impact” on the UK’s on red squirrels, according to the Woodland Trust, but had thrived in the country for over 10,000 years. There are now thought to be fewer than 300,000 red squirrels left, mostly living in pockets of North Wales and Scotland.
Greys, which are on the IUCN’s international list of “100 worst invasive non-native species” compete with red squirrels for food, but they also carry a virus known as squirrelpox.
While the greys are immune to the disease, they transmit it to reds, for whom it is fatal.
On Tuesday, the Department for Environment, Food and Rural Affairs (Defra) told BBC News: "We hope advances in science can safely help our nature to thrive, including through the humane control of invasive species."
A coalition of forestry and conservation organisations named the UK Squirrel Accord (UKSA) is leading the proposals, which are designed to reduce numbers of grey squirrels in the UK.
The organisation has said a “species-specific delivery hopper”, is being designed to administer the spiked food, and modelling work is being carried out “to better understand the potential impact of fertility control on grey squirrel populations in the landscape”.
In a blog post the organisation said grey squirrels are known to cause significant damage through bark-stripping broadleaf trees, such as oak, exposing timber to fungal and insect attack and impacting woodlands’ reproductive resilience.
The plan to use contraceptives to control grey squirrel numbers could be used alongside the existing culls which are carried out.
“Ecological modelling suggests that fertility control alone is unlikely to achieve a rapid reduction in grey squirrel populations," the Squirrel Accord said.
“However, when applied to the low-density populations following short-term culling, eradication could be achieved within the same timescales as continuous culling alone but with substantially lower costs.”
The plan also has backing from Prince Charles, who is a huge fan of red squirrels, to the extent that he has on occasion encouraged them into the royal household in Scotland.
Writing last week in his capacity as patron of the Red Squirrel Survival Trust, the prince said: “These charming and intelligent creatures never fail to delight.
“I take enormous pleasure in having them around – and in! – the house when I am at home in Scotland. They are such inquisitive and delightful characters.”
The Royal Forestry Society, which is a member of the Squirrel Accord is backing the plan.
The organisation said: “We have called for the government to make research and effective control a priority for forestry. Without a sustained programme to control grey squirrels, some landowners are turning their backs on broadleaved planting.”
On their website the organisation added: “In an oak plantation in East Anglia, £10,000 a year is spent on squirrel control – working out at £58 per hectare. If replicated across all the broadleaved woodland in England and Wales that would cost more than £65m.”
The UK Squirrel Accord has tested the feeding apparatus in East Yorkshire to check the contraceptive-laced food can only be accessed by squirrels. The trial indicated up to 90 per cent of the grey squirrel population being studied visited the traps.
A Defra spokesperson told The Independent: “Invasive non-native species such as grey squirrel and muntjac deer threaten our native biodiversity and cost the economy £1.8bn a year.
“The government is committed to ensuring the wide ranging impacts of invasive non-native species are reduced, and the Invasive Alien Species (Enforcement and Permitting) Order which came into force in December 2019 is an important tool to achieve this aim.”