Sir Jeremy Heywood is reported to have told Prime Minister David Cameron about the conversation.
"Our clear impression was that he was spitting tacks with Gove and was ready to resign unless the Minister was told to shut up," a Government source told the newspaper.
Mr Gove, a former journalist at the News Corporation-owned Times, told a Parliamentary Press Gallery lunch in February that he saw "dangers" in the inquiry into press cultures, practices and ethics.
Lord Justice Leveson instructed his officials to compile a full report of the comments and he phoned Sir Jeremy within 24 hours, it was reported.
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