Private-school leaders are warning that attempts to abolish their schools will prove to be a "vote-loser".
Labour has backed plans for private schools to lose tax benefits and be "integrated" into the state sector.
But Fiona Boulton, chairwoman of the Headmasters' and Headmistresses' Conference (HMC), said that shutting down private schools was "based on ignorance and the desire to damage".
Labour said children's life chances should not depend on parental wealth.
The challenge from the HMC comes as it begins its annual conference in London. It follows Labour's decision to close down the independent school sector - including calls for their assets to be "redistributed".
There has been an angry reaction from private-school leaders - and Mrs Boulton, head of Guildford High School, will tell the conference that Labour has misjudged the public mood.
The HMC will publish its own survey findings, a ComRes poll of more than 2,000 people in England, Wales and Scotland, showing 68% support the principle of parents being able to pay for private education.
A Labour spokesman rejected the criticism from the private schools, saying: "Of course the establishment will try to defend its privileges."
But the party appears to have pulled back from suggestions of seizing private school assets. Shadow chancellor John McDonnell last week played down what he called "draconian measures".
Independent schools have argued that if they were put out of business, it would mean state schools would have to find funding and places for those extra pupils.
But shadow education secretary Angela Rayner has told private school leaders that a Labour government's first budget would "immediately end the tax breaks gifted to private schools".
