The 21-year-old suspect has been named as Yahyah Farroukh.
Following the blast Prime Minister Theresa May raised the United Kingdom terror threat level to "critical" - meaning that an attack is expected imminently. They lowered it Sunday to "severe", and police said the investigation was making rapid progress.
"It is inevitable that so-called Islamic State, or Daesh, will reach in and claim responsibility", she told the BBC, using an Arabic acronym for the group.
A 48-year-old man and a 30-year old-man were arrested by the Metropolitan Police Counter Terrorism Command in connection with Welsh law enforcement officials.
According to a Facebook profile thought to belong to Farroukh, he is originally from Damascus, in Syria, and studied English for speakers of other languages at West Thames college, the Guardian reported. Rudd said authorities will try to determine how the suspects may have been radicalized.
Police have searched several properties, including a home in Sunbury where an elderly couple served as foster parents for many children over several decades.
Police have identified 121 witnesses to the attack at the Parsons Green station on Friday and spoken to 100 of them already.
Assistant commissioner Mark Rowley, national lead for counter terrorism policing, said: "Members of the public will be wondering what that means in terms of their travel into work [on Monday]".
On Monday, five US 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.
Most of the injured in Friday's explosion aboard a District Line train suffered flash burns while some were injured in the panicked rush to leave the Parsons Green station where the blast occurred. So the public will still see that high level of policing presence - some armed, some unarmed. Officials say the injuries would have been far worse if it had fully detonated.
Four other violent attacks in Britain this year have killed 36 people.
Earlier this year, assailants with vans and knives attacked passers by on Westminster Bridge and London Bridge in two separate strikes, and a van was driven into worshippers outside a mosque in Finsbury Park.