Residents will be given £500 in cash followed by a bank payment for the rest from Monday.
The Prime Minister insisted the Government was doing everything possible to help those caught up in the tragedy.
The money will come from the £5 million fund announced by Mrs May on Friday.
No 10 said the £500 cash payment is already being handed out and further payments will be available from the Westway Centre and the nearby post office in Portobello Road.
Help will be given to residents who do not have bank accounts.
Mrs May said: “As we continue to respond to the needs of the community, our focus is on ensuring that all of those affected by this unimaginable tragedy get the right support as quickly as possible.
“My government will continue to do absolutely everything possible to help all of those affected through the difficult days, weeks, months and years ahead.”
More to follow