The death toll from the fire is now believed to be around 80, the vast majority of whom were from just 23 flats.
Prime Minister Theresa May has insisted residents will be given a say over the direction of the investigation and Sir Martin’s appointment is expected to be officially announced on Thursday.
As a lawyer, Sir Martin specialised in commercial law before spending more than 20 years as a judge of the Commercial Court and Court of Appeal.
Survivors and the families of those who died in the devastating blaze will be given state funding for legal representation at the probe.
The Prime Minister said a full judge-led inquiry was needed to ensure the events at the 24-storey block were “properly investigated”.
The Metropolitan Police has said it will be months before the final roll call of victims is established.
Detective Superintendent Fiona McCormack said contact had been made with at least one occupant from 106 of the 129 flats in the building.
From those properties, 18 people are dead or assumed dead, meaning the remaining victims are thought to have been in the flats wiped out by the inferno.
She said: “There are 23 flats that, despite huge investigative efforts, we have been unable to trace anyone that lives there.
“At this stage we must presume that no-one in those flats survived, that includes anyone who lived there or was visiting them.”
Pressed on whether the death toll could climb to triple figures, Ms McCormack said: “I’ve said I believe it could increase, I don’t think it’s going to be triple, no.”
A six-month-old baby was among the latest victims to be identified, found dead in her mother’s arms in the smoke-filled stairwell.
Westminster Coroner’s Court opened and adjourned inquests into seven Grenfell Tower victims, including baby Leena Belkadi and her eight-year-old sister Malak.
As the vast criminal investigation into the fire continues, 60 organisations have been identified as having a hand in the tower’s refurbishment, which is suspected to have helped the blaze spread.
Mrs May confirmed on Wednesday that 120 tower blocks across 37 local authority areas had flammable cladding on their exterior.
The National Housing Federation has called on the Government to stop its testing of cladding and instead focus on making people safe.
Chief executive David Orr said: “These tests were the right thing to do, but the results are now conclusive: ACM cladding simply does not pass these tests and is deemed unsafe.
“Across the country, valuable resources - from specialist equipment to expert time - are being poured into a testing process of which the results are already known.
“We are calling on the Government to halt the testing on ACM cladding and shift its focus to making people safe.”
Mr Orr spoke of the testing process revealing a “systematic failure” around the development, manufacture and regulation of cladding.
He said: “It is highly likely that (in order to make people safe) this means removing the cladding from hundreds of buildings we were assured were safe - including hotels, private sector blocks, hospitals and student accommodation.
“This process has powerfully demonstrated a systemic failure in construction, manufacturing and the way that regulation has been applied.
“Putting this right will require a strategic look at the issue of fire safety in buildings as a whole, as well as clear prioritisation and funding from government - the costs should not fall on the residents of these properties.”