In an apparent reference to comments the prime minister has been criticised for in the past, the Labour Party leader said a failure of leadership is “when you use racist remarks to describe people in different countries or in our society”.
He added: “I will never do that and my party will never do that.”
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Mr Corbyn was retaliating to a comment made by Mr Johnson during a BBC election debate on Friday evening, in which he said the Labour politician’s “unwillingness to stand up for Jewish people in the Labour Party” was in his view “a failure of leadership”.
Responding to a question about Islamophobia and antisemitism, the Conservative leader said: “I think that, you know, Mr Corbyn, I’m sure is very well intentioned.
“But I think in his handling of this particular issue [antisemitism], his unwillingness to take a stand, to stand up for Jewish people in the Labour Party, his unwillingness to protect them, to put an arm round them, is in my view a failure of leadership.”
Mr Corbyn said he hoped the prime minister would understand the importance of everyone “using respectful language of all faiths, all religions and all languages” in a multi-cultural society.
It was put to Mr Corbyn that it was “extraordinary” that the Chief Rabbi claimed he was “unfit for office” during the election campaign.
Mr Corbyn replied: “Antisemitism is wrong and totally unacceptable in our society, in our parties, in our political life, and the history of the awful treatment of the Jewish people in the 19th century in Germany indicates exactly where that kind of language and that kind of racism leads us to.
“There is no place for it whatsoever anywhere in our party or our society and there never will be as far as I’m concerned.”
Mr Corbyn added: “I do not ever use racist language in any form to describe anybody in this world, or in our society.”
Challenged on Tory candidates retweeting far-right activist Tommy Robinson, Mr Johnson replied: “All those candidates have either apologised or are now subject to investigation.”
In past newspaper columns, Mr Johnson has made comments about “piccaninnies” with “watermelon smiles” in reference to black people, while also suggesting Muslim women wearing niqabs looking like “letterboxes”.