Eighty-seven tonnes of goods offered following the fire more than four weeks ago are still to be sifted through, the BBC reported.
In total around 174 tonnes of items were donated to help survivors after the blaze, the British Red Cross said.
The challenge in dealing with the masses of donations is “really off the scale”, Diana Goss from the charity told the broadcaster.
The British Red Cross said 10 tonnes - less than 6% - of items have been or will soon be made available to former residents of the burnt-out tower block.
Other donations are being sold in charity shops or to recycling companies, with the proceeds returned to victims’ funds.
Several millions of pounds in donations have been collected by charities and other groups since the tragedy, which is thought to have claimed the lives of at least 80 people and displaced hundreds more.
Of the estimated £20 million in private donations made after the fire, less than £800,000 has been given out, the BBC said.
Last week Grenfell community leaders demanded to know how money collected for victims of the blaze will be distributed, with some volunteers and residents saying victims had not been consulted on how it would be spent.