The man was arrested before dawn by counter-terrorism officers in the Welsh capital Cardiff and has been taken to a police station in London for questioning.
Police did not name the suspect but said he was arrested under the Terrorism Act.
An 18-year-old man, Ahmed Hassan, was charged on Friday with attempted murder and the use of explosives.
Two other men, aged 25 and 30, who were arrested in Newport in Wales last week, are still being questioned.
Placed on a packed train during morning rush hour, the improvised device injured 30 people when it went off despite apparently malfunctioning.
The attack in Parsons Green, a quiet residential area, caused panic among commuters and some were hurt in a stampede as people fled the scene.
The Tube bombing was the fifth terror attack in Britain in sixth months — four of them in London and one in Manchester — with the bloodshed claiming 35 lives.
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