The Sunday Times reported Mr Milne told party officials to reinstate Mr Secker after he was suspended for joining a Facebook group that contained Holocaust denial.
Mr Murray said Mr Corbyn was “interested in this one”.
Lord Falconer warned: “Us being characterised as antisemitic is an existential threat to the Labour Party.
“I think it is such a serious matter, if we don’t deal with it as time goes on we will be eroded as a political force in this country.
“How can you ask people to vote for a party that was antisemitic? It goes absolutely against the essential values of the Labour Party.
“The Jewish people I’ve been speaking to have been saying this is an existential threat to us - the idea that the prospective Prime Minister and leader of the opposition and his party makes us feel completely unsafe."
Lord Falconer, who was Tony Blair’s justice secretary, disputed claims that his membership of the party barred him from carrying out any future role leading an antisemitism inquiry.
He said he had been granted a request for accesss to Party emails and to his “own team of lawyers” but had yet to nail down “additional resources” required.
Responding to claims from Dame Margaret Hodge that he had bullied her last year after she faced disciplinary proceedings for labelling Mr Corbyn an “antisemite”, Lord Falconer said he was trying to do his best to stop proceedings being brought against the veteran Jewish MP.
