MPs challenged the Labour leadership last month over the case of Kayla Bibby, who posted an antisemitic image of an alien with the Star of David on its back grasping the Statue of Liberty by the face, suggesting control over the US.
The 33-year-old activist added the Facebook caption: "The most accurate photo I've seen all year!"
The case was examined last year and Ms Bibby, who attended Labour's conference in September as a delegate from the Liverpool Riverside constituency, was let off with a warning.
Emails leaked to The Times on Tuesday revealed that "an ally of Mr Corbyn" rejected the suggestion by Labour Party staff to suspend her. They said that the images were not antisemitic.
Mr Gardiner, a councillor on London's Camden Council, is believed to have overseen numerous other complaints about alleged antisemitism over the past year. There were claims earlier this week he had resigned from his job - but it was then suggested by Labour sources he was on holiday.
