Moore was dismissed from HBOS in the run-up to the financial crisis in 2004 for warning that the bank was running risks it did not understand.
He said: "Staley should be fired. Trying to find out the identity of an anonymous whistle-blower where the motivation is obviously to try to crush them is gross misconduct."
Moore added: "It wasn't just once. He did it again after a warning. The rules have been clear for a very, very long time."
The remarks will pile pressure on Staley, who is facing four separate regulatory probes on both sides of the Atlantic.
Read also: Barclays CEO in probe
The Prudential Regulation Authority and the Financial Conduct Authority are looking into the matter in Britain, while New York’s Department of Financial Services and the US Department of Justice are conducting investigations in the US.
Staley enrolled cyber-experts employed by Barclays to try to discover the identity of the author of a letter highlighting personal issues relating to Tim Main, an investment banker who had previously worked with Staley at JPMorgan.
Labour MP John Mann said Staley should resign. Liberal Democrat leader Tim Farron said the Barclays boss would have to go if official UK investigators found evidence of "serious wrongdoing".
Barclays said it had given Staley a formal written warning and would slash his salary. The bank is said to be desperate to hold on to him and has said it believed he made an honest mistake in thinking it was permissible to try to identify the letter writer.
Staley’s internal security team had tried to enrol the help of the US Postal Service to identify the whistle-blower. A security source said: "The big retail banks have moved from having hundreds of security staff to thousands".
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