The letter says the LRC are "condoning and nurturing this racism" and says LAW and JVL "ridicule and minimise the real lived experiences of victims of antisemitism".
JLM says the LRC's opposition to the International Holocaust Remembrance Association's definition of antisemitism is "in no way consistent with any established conventions within anti-racism movements".
The letter says: "Such a culture feeds an environment whereby Jewish MPs such as Luciana Berger receive death threats, abuse and threats of violence. It feeds a culture whereby Jews across the party are targeted for deselection, sanction or intimidation.
"It feeds a culture of not believing Jews when they are on the receiving end of hate. It feeds the cancer of antisemitism in our party."
The JLM says supporting LRC without challenging it on antisemitism would be inconsistent with the line Mr McDonnell took earlier this week, when he challenged a phone-in caller to a radio show who told him accusations of Labour antisemitism were "smears".
"LRC are feeding the problem and unless they change their position of condoning and nurturing this racism in our party then you cannot and should not have anything further to do with them and they also should have no place in the Labour Party," JLM's letter says.
"We look forward to hearing your comments this weekend."
Mr McDonnell is due to speak on Saturday morning. A Q&A will follow. He is running unopposed for re-election to his position.
