Edinburgh Council said it has 12,805 spare places amid Conservative claims the tax policy could lead to an exodus and state schools being overrun.
Labour leader Keir Starmer has said 20 per cent VAT will be added to school fees if he wins power.
The revenues will be used to hire new teachers for comprehensives, with Holyrood landing an extra £150million.
But critics of the policy claim the tax will be unaffordable to parents on modest means who go private.
They claim thousands of pupils will leave private education, piling even greater pressure on a struggling state sector.
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