The development follows last week’s public row over the Mail’s description of Ralph Miliband as "the man who hated Britain".
In yesterday’s Mail editorial, references to Ralph Miliband were wrapped up in an attack on The Guardian as "the paper that helps Britain’s enemies".
The Mail concluded the editorial thus:
"As for [The Guardian’s] attack on us over the Labour leader’s father, let us say something in Ralph Miliband’s favour.
"True, he hated so much about this country that he wanted a workers’ revolution to overturn everything from the monarchy to Parliament, property rights and the common law. And, yes, his ideas chimed more with Stalin’s than with Churchill’s during the Cold War.
"But he fought for Britain in the war. And never once, as far as we are aware, did he give practical help to our enemies.
"Nor was he ever accused by the head of our security services of putting British lives at risk.
"Isn’t that a great deal more than can be said for The Guardian?"
The senior party source described the editorial as a "significant step by the Daily Mail".