A consultation period with victims and other parties about the scope of the probe is under way and is expected to wrap up by the parliamentary recess.
This gives survivors concerned the process will be too “narrow” until July 20 to make their case to Sir Martin Moore-Bick that its parameters should widen.
Discontent has been brewing after the judge leading the inquiry suggested it will largely focus on the cause of the fire and how it could be prevented in future.
Campaigners warned a boycott could be afoot unless the systemic issues underlying the blaze, in which at least 80 people have died, becomes a central plank of concern.
An inquiry spokesman said: “The aim is to have (the terms of reference) done before Parliament rises on the 20th.”
Sir Martin met with survivors and those displaced from homes nearby on his first day in the role and will hold further meetings with other groups in the next week.
Labour MP David Lammy called for the retired Court of Appeal judge to forge closer links with victims so the process could maintain legitimacy in their eyes.
“He is a white, upper-middle class man who I suspect has never, ever visited a tower block housing estate and certainly hasn’t slept the night on the 20th floor of one,” he told Sky News’s Sophy Ridge On Sunday.
“I hope he would do that in the days ahead.
“The job is not just to be independent and judicious - I am sure he is eminently legally qualified, of course he is - it is also to be empathetic and walk with these people on this journey.
“He needs to get close to those victims and survivors very, very quickly and establish he is after the truth and he is fearless and independent and won’t be swayed because he is part of the establishment.”
Yvette Williams, one of the organisers of the Justice 4 Grenfell campaign group, had earlier suggested their co-operation with the inquiry hung in the balance.
She told Sky News: “They cannot just look at 14 June, when that building became an inferno. They can’t do that.
“If we don’t get good terms of reference for the public inquiry and we don’t get a wide remit so that those people can take responsibility for what they’ve done, then we won’t participate in it.”
Meanwhile, the future of troubled Kensington and Chelsea Council was also called into question as one of its councillors suggested the crisis could spell “the end” of the authority.
Leader of the opposition Labour group Robert Atkinson, who was locked in a heated confrontation with leader Nicholas Paget-Brown last week, claimed ties with the community could be irreparable.
Both Mr Paget-Brown and deputy Rock Feilding-Mellen quit their roles amid ongoing criticism of the council’s handling of the disaster.
Mr Atkinson, who represents the ward in which Grenfell Tower is based, told the Press Association: “I really think it could quite quickly be the end of the council altogether.
“I am not sure they have got the capacity or the will to rebuild themselves as an independent authority.”
One minister also questioned whether the council could be scrapped altogether to make way for a larger body covering more areas.
Foreign Office minister Mark Field, a former Kensington councillor, suggested the move could make them better equipped to respond to major crises.
He told the Sunday Times: “I doubt that K&C, or any London authority, has the critical mass to deal with something like this.
“It raises questions about whether the model of London governance, with 33 unitary authorities, is really a sustainable model. Thirty-three does seem too many.
“You could look at the New York model, where they have five boroughs. I think that’s what we will move towards.”