Sir Martin Moore-Bick sent the letter late on Thursday following weeks of discussion with survivors.
There has been controversy over Sir Martin's suitability for the role and the areas the inquiry will cover.
The prime minister will set out the terms of reference for the process in the coming weeks.
Downing Street said: "The prime minister has received the letter and will consider it and respond shortly."
How far the inquiry can go has been a source of controversy since the Sir Martin's appointment was announced.
He angered survivors on his first day in the job by indicating the investigation would be "pretty well limited" to examining the cause of the fire, how it spread and how to prevent future incidents.
But he later vowed to consider a "broad range of evidence" which could include why residents' warnings about fire safety were allegedly ignored by authorities.
Survivors, bereaved families and other involved parties spent weeks making their case about what questions the inquiry should answer.
More than 400 submissions were received during the consultation period.
The "Justice 4 Grenfell" campaign said earlier this month it wanted an examination of local and national social housing policy and whether it "increased risks to residents".
Questions were raised in the aftermath of the disaster about the cladding used on Grenfell and other buildings. More than 100 buildings have failed the latest fire safety tests set in the wake of the fire.
Sir Martin has previously faced calls to resign from residents, while Labour MP David Lammy said he was a "white, upper-middle class man" who had "never" visited a tower block housing estate and should not have been appointed.
The inquiry is due to start in September. Sir Martin has previously said an interim report could be produced within a year.
The prime minister will set out the terms of reference for the process in the coming weeks.
The judge leading the inquiry into the Grenfell Tower fire has written to the prime minister with recommendations for its terms of reference.