More than 100 offers of long-term accommodation have been made to survivors of the Grenfell Tower blaze.
Barbara Brownlee, of the Grenfell Fire Response Team, said the group was on course to have offered accommodation to every family "who want us to make an offer" by the end of Monday.
Eight offers have been accepted so far, she said, adding that families were under no pressure to take up the first property they were shown.
Ms Brownlee, who is director of housing and regeneration at Westminster City Council, told the Press Association: "We've made 126 (offers) as we sit here now.
"We've got somewhere in the region of 10 or 12 to make today and then we will have covered every family who we can make an offer to, who wants to talk to us at this time.
"You will understand completely there are some families who do not want to move yet into the area of free housing.
"They are far too distressed, they are dealing with children still in hospital, or they are burying their relatives, and the last thing they want to do is talk to us about an offer of housing.
"So of those who want us to make an offer and want to engage with us, we will have completed that by the end of play today and we're very confident of that."
Council-owned properties were being offered for temporary or permanent tenancy according to the wishes of residents, while privately managed accommodation is being offered for 12 months, she added.
All the families will have their rent payments suspended for a year from the date they move in.
Ms Brownlee was speaking in a second floor flat in a five storey terraced townhouse in the heart of Kensington and Chelsea, which has just been viewed by a family.
The spacious two-bed, which is managed by Kensington and Chelsea Tenancy Management Organisation, will cost around £1,103 to rent per month and is free to move into immediately.
Ms Brownlee said the value of the Elm Park Gardens property was around £1.7 million to buy.
Furniture, bed linen and crockery will be provided to families to help them make the transition to more permanent accommodation, she said.
Ms Brownlee promised families were "not ever going to be abandoned" by the response team - not now nor in the months after they take up long-term accommodation.
Asked if she had experienced residents saying they did not want to move into KCTMO-managed accommodation, she said she had not.
She said: "Everyone would like to live on a ground floor, and that's pretty challenging in the middle of London, but as low-rise as possible.
"We have a lot of people who don't want anyone above them because it was quite horrendous knowing what was going on above them in the block and they don't want to ever suffer that again.
"So we have a lot of requests about type and location of property but we haven't heard anything specific about whose property it has to be."
The response team is not working toward a specific date by which they will have got all families into more permanent accommodation "because we genuinely don't want to pressure people", she said.
"There's absolutely no pressure on anybody. We are listening to people's stories," she added.
Of the decision to ensure no resident will pay rent for 12 months, Ms Brownlee said: "I think it will take away a lot of worries.
"People are dealing with the most enormous amount of loss, stress and anxiety.
"People are dealing with children who are desperately anxious after what they've been through, they're dealing with elderly parents in hotel rooms that are not the best place to look after older people.
"So I think just to be able to take away one practical worry is a really positive step."