Nine offers of permanent accommodation have been accepted by Grenfell survivors whose homes were destroyed, while one household has already moved in to a new home.
The leader of Kensington and Chelsea Council, Elizabeth Campbell, said she hoped more of the 180-plus households in hotels and temporary accommodation would be rehoused as soon as possible, but according to "their need, not speed".
And she pledged to spend "every penny" of the council's vast reserves on rehousing those affected by the tragedy if necessary, with £76 million set to be made available in a leadership meeting on Thursday.
Some £30 million has already been allocated by the council for rehousing, Ms Campbell said, while it is costing the council £1,000 per household per week to keep families in temporary accommodation.
She said: "It's a lot of money, tens of millions of pounds, but that's what that money is there for.
"We've built up reserves and we built them up in case of emergencies, and this is an emergency.
"And we will spend that. And that's right that we should do it - these are our neighbours.
"If you don't spend money on your neighbours in an emergency, then when do you spend it?"
She was speaking at one of the flats earmarked for Grenfell survivors on Hortensia Road - an hour away by foot (about three miles) from the charred tower block where they once lived.
The other two locations are Kensington Row, with 68 units (about 1.7 miles from the tower) and Tavistock Crescent with six properties (0.9 miles away), while one other house is also available.
They make up a total of 105 properties, with the council planning to buy more from registered housing providers and on the private market.
Since August 21, survivors have been able to browse brochures and register their interest in permanent homes in the three locations on a website set up by the council.
Some 134 people have logged in to the website and expressed an interest within the first three days, said Ms Campbell.
At least 20 permanent offers have been made, and 31 viewings conducted, the council said.
One of the 23 households in temporary accommodation, initially accepted on a 12-month basis, decided they wanted to make their new home permanent, it added.
Ms Campbell said she was "really pleased" with feedback from residents and denied they were being made to bid against each other, saying each home would only be viewed and offered to one household at a time.
She said: "People aren't bidding, they are saying what their preference is.
"We won't show anyone a flat if they can't come in and say 'I want this flat' and we say 'sign up'."
She added: "Absolutely every single person will be phoned up, we will engage one-to-one and we will do it with them."
People will have until September 1 to register their preference on the website, but Ms Campbell said those who felt unable to engage would be pro-actively approached and helped.
Ms Campbell said she would consider "all options" for future management of the estate, after the Prime Minister told residents the tenant management organisation (TMO) would stop overseeing it.
Theresa May acknowledged concerns about the culture of the council, the pace of its response following the fire and the TMO during a meeting with residents on Tuesday evening.
Elizabeth Campbell said she had written to the TMO seeking a "swift transition" of responsibilities for the Lancaster West estate to the council and promised to consult with residents about future plans.
She said: "We wrote to the TMO yesterday and said that we didn't think the most viable option was to work with them going forward."
Asked if she would consider introducing a managing structure where residents had a central part, she said: "I think we will consider all options. The only option we won't consider is the TMO continuing."