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The 25-year-old was arrested in Newport, south Wales, on Tuesday evening, Scotland Yard said. An address in Newport is being searched, reports BBC.
On Saturday an 18-year-old man, who the BBC has learnt had previously been referred to an anti-extremist programme, was arrested in Dover port.
A 21-year-old man was also arrested in Hounslow, west London. Both are still in custody.
A homemade bomb partially exploded in a train at Parsons Green station, injuring 30 people in rush hour.
The 25-year-old was arrested under section 41 of the Terrorism Act.
The arrest was carried out by the Met's Counter-Terrorism Command, supported by Gwent Police and the Welsh Extremism and Counter-Terrorism Unit.
Metropolitan Police Commander Dean Haydon said: "This continues to be a fast-moving investigation.
A significant amount of activity has taken place since the attack on Friday."
"We now have three men in custody and searches are continuing at four addresses."
At 08:20 BST on Friday a homemade bomb, which was transported in a Lidl bag, partially exploded in a Tube train at Parsons Green station, causing burns to a number of victims.