Gavron, Livingstone's deputy and whose parents survived the Holocaust, told BBC One's 'Politics Show' that while she condemns any suggestion Livingstone is anti-Jewish, she conceded the comments made to the Evening Standard journalist were "inappropriate".
She added: "I believe he will come to the point where he says, 'I regret that I have caused offence to the wider Jewish community'."
Livingstone has been under increasing pressure from, among others, Prime Minister Tony Blair and culture and media secretary Tessa Jowell to apologise after making the comments, which were subsequently splashed on the Evening Standard.
Livingstone says the comments were made in reference to Evening Standard sister paper The Daily Mail's pro-fascist stance during the 1930s.
The row has marked a low point in an increasingly acrimonious relationship between Livingstone and Associated Newspapers' Evening Standard under the editorship of Veronica Wadley.
This included a row about the congestion charge in which Livingstone described one headline as a "bare-faced lie", in a letter printed in the paper.
Last month, he opened the way for Richard Desmond to launch a freesheet rival to Associated's Metro.
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