The six were taken to a south-east London police station, the force spokesman said.
A 29-year-old man was arrested in the street in west London, a 21-year-old man and the women were arrested at a separate residential premises in west London, and three other men, aged 18, 24, and 26, were arrested in east London.
Searches under the Terrorism Act 2000 are being carried out at eight residential premises in east, west and north London and one business premises in east London.
Public safety remains our overriding concern.
More top news
"I do not accept the allegation that I have brought the Labour Party into disrepute", said Livingstone.
Family and friends of the victims of Grenfell Tower have been paying tribute to those who perished in the disaster.
The balmy conditions have swept in on a high pressure system, dragging in hot air from the continent, according to the Met Office.