"The real fundamental difficulty is that we are finding that people are reluctant to come forward, either due to their immigration status or because of the basis on which they were either staying or living in the tower", Victoria Vasey, director of the local North Kensington Law Centre, told the BBC.
The mayor of Greater Manchester, Andy Burnham, said on Tuesday that he had met with Greater Manchester fire and rescue service, social landlords and local authorities from across the region to agree on an action plan to reassure people.
UK PRIME MINISTER Theresa May has apologised for the official State response to the Grenfell Tower fire believed to have killed 79 people, saying it was "just not good enough". "As Prime Minister, I apologise for that failure and as Prime Minister I have taken responsibility for doing what we can to put things right".
Demonstrators march in central London, many demanding justice justice for the victims of the recent deadly apartment block fire at Grenfell Tower, as they march towards parliament in central London Wednesday June 21, 2017.
Kunwardia said: "I've already spoken to a number of former Grenfell Tower residents who have refused the offer of temporary accommodation outside of the borough and have opted to stay with friends and family instead".
"The prime minister is very aware that people want answers promptly and we want to get this going promptly, so that's our intention", the spokeswoman said. Dozens of people died in the fire, which gutted the 24-story residence and left many people without homes.
Seventy-nine people have been confirmed as dead or missing presumed dead after the blaze which started in the early hours of 14 June.
However, they did say: "Four of our blocks do have cladding (Oldmead House, Bartlett House, Mersea House and Colne House), but this is not the same type as the cladding used on Grenfell Tower".
"Until we actually know what went wrong at Grenfell Tower, the more we investigate the better", Malhotra added.
John Barradell, head of the newly established Grenfell Fire Response Team (GFRT), admitted that even a week after the blaze, the effort still needed to be stepped up to help victims.
The new housing move comes after Labour leader Jeremy Corbyn called for luxury empty properties in Kensington to be taken over by the government to house victims, a demand rejected by ministers.
Since the tragedy, the Government announced it was triggering a £5 million emergency aid fund to help those affected.