The BBC has been told there will be a concerted push for Labour to adopt an independent process for dealing with complaints of anti-Semitism.
A group will make the demand at Monday's regular meeting of MPs.
But political correspondent Iain Watson said it would also, crucially, be made at the next meeting of Labour's ruling national executive later this month.
Emily Thornberry said earlier Labour must heed "the message" on anti-Semitism, not attack the "messengers".
The shadow foreign secretary told the BBC's Andrew Marr "nobody can pretend there isn't an ongoing problem" within the party and with "our processes for dealing with it".
Labour has been engulfed in a long-running dispute over anti-Semitism within its ranks, which has led nine MPs and three peers to leave.
Last week, Panorama revealed claims from a number of former party officials that some of Jeremy Corbyn's closest allies tried to interfere in disciplinary processes involving allegations of anti-Semitism.
Labour's disputes team is supposed to operate independently from the party's political structures, including the leader's office.
The party has insisted the claims are inaccurate and made by "disaffected" former staff.
On Sunday, Jewish Labour MP Dame Margaret Hodge told Sky News the party was at "a tipping point" on the issue, and it was "intolerable and unacceptable" for the leadership to attack those who went public with concerns.
"If the leadership doesn't start to listen now there will be many more people who will feel so uncomfortable within the Labour Party that they can no longer remain," she said.
Dame Margaret also said Labour must urgently adopt "a totally independent complaints mechanism, because it is clear that the current mechanism has been abused, it has been abused by political influence".
That call was echoed by the Tribune group of Labour MPs - which is on the centre left of the party and includes former frontbenchers.
They urged the National Executive Committee (NEC) to set up an independent investigation into the allegations aired by Panorama, and to create "an independent complaints procedure with representation from the Jewish community, which is totally independent from the leadership of the Labour Party".
The intervention of the Tribune group of MPs is significant.
The group includes Yvette Cooper, a minister in the last Labour government.
But many of the other members are to the left of the old Blair/Brown axis - and include Ed Miliband's former chief of staff Lucy Powell.
They feel that if the party is to detoxify the anti-Semitism row then it's not enough for the complaints system to be independent of the leadership, it must be seen to be independent.
They want to see allegations of political interference in the current system investigated, too.
In the past, when a more independent system was mooted, senior staff were concerned about data protection and of 'outsourcing' investigations to a new body which may have little practical experience of Labour's rules.
But as the corrosive row continues, a wider group than just the Tribune MPs are privately calling for more radical action.
In fairness, the Labour leadership feel they are getting very little credit for pouring more resources in to the existing system and speeding up the complaints process.
But it will be for the party's ruling body - the NEC - to decide whether a whole new approach should now be attempted.
Earlier, Ms Thornberry said she found the Panorama episode "awful" - both the programme itself and "more importantly the revelations".
She said she understood the party had concerns about how the investigation was conducted, but "the message.... is what is important".
In May, the Equality and Human Rights Commission (EHRC) launched a formal investigation into whether Labour had "unlawfully discriminated against, harassed or victimised people because they are Jewish".
Ms Thornberry said it was "a shame and a disgrace" that the EHRC had been brought in "but they have and we should welcome it".
She said the party needed to set up a complaints process "that is tough and that works and is an example of good practice", and should be asking the commission: "Can you help us?"
'Sorting it out'
Shadow chancellor John McDonnell told BBC Radio 4's The World This Weekend things were improving.
"I think we are sorting it out actually. I think we've got a new general secretary, we've put in place a system now," he said.
"There's always lessons to be learnt, but I think the way Jennie Formby, our general secretary, has operated - implemented the measures - is getting on top of this."
Labour has rejected claims of interference in its disciplinary processes and described the Panorama programme as "seriously inaccurate" and "politically one-sided".
In a statement, it said: "Since Jennie Formby became general secretary the rate at which anti-Semitism cases have been dealt with has increased more than four-fold.
"We will build on the improvements to our procedures made under Jennie Formby, and continue to act against this repugnant form of racism."