The Jewish Voice For Labour member also repeatedly joked about what she called “Jew process” at activist meetings and claimed it had been wrong to expel Marc Wadsworth from Labour after his confrontation with Jewish MP Ruth Smeeth.
In a statement on Friday, the JLC and the Merseyside Jewish Representative Council said her potential candidacy was "profoundly concerning to the mainstream Jewish community".
"She has previously been suspended from the party for indiscretions and comments that were designed to minimise racism against the Jewish community," they said.
"The fact she is under consideration for selection emphasises why the Equalities and Human Rights Commission are investigating the Labour Party for institutional antisemitism."
"If the Labour Party are serious about dealing with the issue of anti-Jewish racism, they must consider whether it is appropriate for Cllr Bird to be on the shortlist for a Parliamentary seat."
The selection was prompted by Dame Louise's decision last week to leave Labour.
She said the party, of which she had been a member for 55 years, was "no longer a safe place for Jews" and Mr Corbyn "must bare the responsibility for this".
