Two other men arrested over the weekend - an 18-year-old refugee from Iraq and a 21-year-old from Syria - remain in custody.
Following the 7/7 bombings in 2005, the government attempted to get the maximum period of detention without charge extended to 90 days, but the proposal was defeated in the House of Commons.
The UK's terror threat level has been lowered from "critical" to "severe", Home Secretary Amber Rudd has announced.
Counter-terrorism officers have arrested two men after a crude homemade bomb sent flames shooting through a packed carriage at west London's Parsons Green underground station during the Friday morning rush hour, injuring 30 people although it apparently failed to detonate fully.
On Saturday, an 18-year-old man was detained at Dover port and a 21-year-old was arrested in Hounslow in west London.
The 18-year-old, held at Dover port, is believed to be an Iraqi refugee orphan who moved to the United Kingdom when he was 15.
Commander Dean Haydon, head of the Met's Counter Terrorism Command, said: "This continues to be a fast-moving investigation".
Mr Ali said the arrested man worked as a painter and decorator for some local residents.
But the very fact that yesterday's attack took place with no warning shows this system is only a broad guide to the threat and simply reflects the latest assessment.
"It is inevitable that so-called Islamic State or Daesh will try to claim responsibility but we have no evidence to suggest that yet", she told the BBC.
On Monday, five USA and British government sources said it was possible that the attack was somehow "inspired" by Islamic State via its extensive array of internet-based propaganda and instruction materials.
The level had been raised on Friday to the highest - critical - following the bucket bomb blast at Parsons Green Tube station that left 30 passengers injured.
A local official said the second suspect had also been fostered by the Joneses.
"It is necessary for us to look, as we are doing, at whether our police and security service have the full capabilities, the powers that they need", May said.
Friday's bombing was the fifth major attack regarded by authorities as a terrorist incident in Britain this year which have claimed the lives of 36 people.