Mr Corbyn has written to Theresa May saying that such a move would increase confidence in the process, and urged her to minimise further suffering of survivors of the blaze.
The Labour leader wrote: "I urge you to consider broadening the inquiry team to a model more similar to that used in the McPherson Inquiry, including with representation from those from minority backgrounds, in order to support the judge leading this inquiry."
Labour wants the probe to be split into two parts in order to "get answers to urgent questions" about what happened at Grenfell Tower.
The first part of the inquiry should report back this summer "to minimise further suffering of survivors", Mr Corbyn said.
The second part of the investigation would take a "wide-ranging" look at the issues thrown up by the disaster, according to Mr Corbyn.
His letter states: "We would be disrespecting the memory of those who died in the Grenfell fire, and putting further lives at risk, if we fail to fully learn these lessons.
"It is therefore our view that an immediate inquiry into the proximate causes of Grenfell should be supplemented by a longer-term, more wide-ranging inquiry into the underlying causes of what went wrong at Grenfell and the extent to which they are replicated on a national scale."
Mr Corbyn said phase one of the inquiry should concentrate on what started the fire, why it spread so rapidly, and whether building regulations were contravened.
It would also focus on why residents' complaints about the condition of the building's fire safety features were not acted upon, the type of advice given to people during the blaze, and whether Kensington and Chelsea's funding and administrative decisions had any impact on the circumstances surrounding the disaster.
The probe should also look at the level of help and advice given to survivors in relation to housing, mental health issues, benefits and immigration status after the incident.
The second part of the inquiry would examine present fire regulations, housing allocation policies, and funding for local councils and the fire service, the Labour leader said.
Mr Corbyn wrote: "There is considerable concern among residents and others that the judge leading the inquiry has already been directed towards narrowly defined terms of reference, which will not bring residents the answers they seek.
"The importance of residents and victims' families having full confidence in this inquiry cannot be under-estimated."
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