Scotland Yard's counter-terrorism squad, SO15, arrived at the scene at Parsons Green Tube station and took the lead in the investigation from the British Transport Police.
London mayor Sadiq Khan said: "The police investigation is ongoing and there will still be significant activity today and over the days ahead. For strong investigative reasons we will not give any more details on the man we arrested at this stage", he said.
The terror group has frequently declared that it was behind terrorist incidents in recent years but experts have warned their claims should be treated with caution.
Britain was hit by a terrorist attack on Friday morning, when a crude device exploded on a crowded London Underground train.
The Philippine Embassy in London has urged Filipinos in the city to remain calm following the attack on a packed London Underground train that injured at least 29 people Friday.
On request from the police, and with authorisation from the Defence Secretary, the first phase of Operation Temperer is being partially enacted.
'Islanders are asked to remain vigilant and to report any suspicious behaviour to the police.
It comes after the arrest of an 18-year-old man in Dover, in the departure area of the port.
The Islamic State in Iraq and Syria (ISIS) said an affiliated unit had carried out the attack.
May said the military would be providing support to police and would replace officers on guard duty at national infrastructure sites that are not accessible to the public.
Following the first arrest, police launched a massive armed search in the southwestern London suburb of Sunbury.