But fellow Labour candidate Jon Cruddas said private schools should lose their charitable status entirely.
Conservatives accused Mr Johnson of making "clumsy threats" over the issue.
Private schools claim many of them would close if not for the annual £100m in tax breaks from being a charity.
New charity rules mean private schools now have to pass a public benefit test showing how they add to communities.
Some teachers said that they found it patronising of Mr Johnson to imply that state school teachers needed help.
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