Mr Miliband's remarks come after Mr Umunna, a rising star in the Labour Party, told the Sunday Times that he had smoked marijuana when he was a teenager hoping to become a DJ. He said he briefly ran a club night in Brixton, south London.
"I've smoked soft drugs. But it's not something I'm proud of and I don't think it's news any more, to be honest," he told the newspaper.
Asked what he had smoked, he said: "Certainly nothing beyond marijuana. That industry is rife with it.
"That was a part of the industry I found unattractive."