But one source told the JC: "Both of these proposals change very little. They do not get to the heart of the problem - we need independence in our disciplinary processes.
"The General Secretary's involvement is certainly not going to guarantee that judging by her previous form."
A Jewish Labour Movement source said of the second option: "The NEC officers are creatures of the leadership. They will do whatever LOTO tells them.
"That is why they voted 4-1 not to see the EHRC submission - even though it was made in their name. They're Unite nodding heads."
Mr Corbyn appeared to repeat past comments about disciplinary processes not being "good enough" at the meeting in Westminster on Monday afternoon.
Jennie Formby, Labour's General Secretary, also updated the shadow cabinet on the complaints process for the first six months of the year.
Her report suggested that eight members had been expelled and three were given extended suspensions.
Another 116 members have been suspended, 100 cases lacked sufficient evidence, 163 showed no rule breaches, while a further 146 were still being processed.
Ms Formby revealed that 97 out of 190 cases of alleged antisemitism had been referred to the party's highest disciplinary body, the National Constitutional Committee.
At present only the NCC has the power to expel members from the party.
