Home Secretary Amber Rudd said police had made “good progress” in the investigation as she announced the country had been removed from the highest terror alert.
Ms Rudd said: “Following the attack in Parsons Green last Friday the police have made good progress with what is an ongoing operation.
“The joint terrorist analysis centre, which reviews the threat level that the UK is under, has decided to lower that level from critical to severe.
“Severe still means that an attack is highly likely so I would urge everybody to continue to be vigilant but not alarmed.”
It comes after police arrested a 21-year-old man in Hounslow at 11.50pm on Saturday. Searches are taking place at a residential address in Stanwell, Surrey, in connection with the arrest, Scotland Yard said.
An 18-year-old man was also detained in the departure area of Dover ferry port on Saturday morning.
The home secretary has said it appears the bomber was not a lone wolf but it is “too early to reach any final conclusions on that”.
Asked if she could give any information on claims by Islamic State that there were other unexploded devices, she told BBC1’s Andrew Marr Show: “We don’t.
“It is inevitable that so-called Islamic State, or Daesh, will reach in and try and claim responsibility. We have no evidence to suggest that yet.”
The country’s top counter-terrorism officer, Assistant Commissioner Mark Rowley, said police were gaining a “greater understanding of the preparation of the device”.
He said: “The high pace and rapid progress of this investigation is continuing.
“There was another arrest overnight and two men are in custody. There are now two searches continuing at addresses in Hounslow and Surrey and we are getting a greater understanding of the preparation of the device.
“There is still much more to do but this greater clarity and this progress has led JTAC - the independent body that assesses threat - to come to the judgement that an attack is no longer imminent.”
Armed police will maintain a strong presence across the country and military personnel drafted in to provide support will be phased out, Mr Rowley said.
On Saturday, a search was carried out at a house in Sunbury-on-Thames, Surrey - home to foster care couple Penelope and Ronald Jones, aged 71 and 88 respectively, who previously received MBEs for services to children and families.
The couple are said to be staying with friends.
Friend Alison Griffiths said the couple had an 18-year-old and a 22-year-old staying with them recently.
She described Mr and Mrs Jones as “great pillars of the community”, adding: “They do a job that not many people do.”
Jim Adaway, a family friend, said the Joneses recently returned to foster caring to help resettle youngsters from overseas.
The 37-year-old said: “They have been here a lot longer than me. All I know is that they gave up the fostering and someone got in touch with them and they started taking kids again, refugees, about a year ago.”
Mrs Jones had been struggling with one of the children in their care, he added.
“I think Penny was getting in touch with (the authorities) saying ‘I cannot handle this one’.”