Two overground alternatives to the tunnel at the Wiltshire monument, due to be outlined today in consultation documents, would destroy nesting and roosting sites of the stone curlew, The Royal Society for the Protection of Birds says.
The stone curlew has two UK strongholds, one of which is the area surrounding the Stonehenge World Heritage Site.
The new road plans would also harm prospects for more than 25 other bird species and at least 14 types of butterfly, the RSPB said.
The tunnel was given the go-ahead by the Government after a public inquiry in 2004, when ministers called it an "exceptional environmental scheme".
But the Highways Agency says the 1.3-mile tunnel will cost an estimated £470m, up from £284m, because of complications that make tunnelling more difficult than envisaged.
Tony Richardson, the director of the RSPB's south-west region, said: "A completely new road through the Stonehenge site is unthinkable, not only because of the area's obvious archaeological value but also because of the public outcry it will spark among the millions of people who value Britain's wildlife.
"It is less than six months since we hit targets for stone curlew recovery both nationally and in the south-west, where one third of the UK's population is found.
"Approval for any overground route will make a mockery of the Government's desire to get this mysterious and vulnerable bird back on its feet."