The mayor urged people to "remain calm and vigilant" in the wake of the explosion, which occurred at around 8:20 a.m. BST on Friday at Parsons Green subway station.
Islamic State has claimed responsibility for the blast, the militant group's Amaq news agency said Friday evening.
The suspect is reportedly being held in a London area police station for questioning.
Parsons Green station has reopened to passengers less than 24 hours after a 'fireball' bomb exploded in a train in the middle of rush hour yesterday.
The president said he planned to call British Prime Minister Theresa May "right now".
She said: "The public will see more armed police on the transport network and on our streets, providing extra protection".
This is the first arrest in connection with the incident. According to Mark Rowley, Britain's highest-ranking counter-terrorism officer, the investigation has been advancing very well.
An underground tube train stands at a platform at Parsons Green underground tube station in west London.
Police said they had spoken to 45 witnesses so far and received 77 images and videos from the public, and urged those with information to get in touch. "Although we are pleased with the progress made, this investigation continues and the threat level remains at critical".
"Another attack in London by a loser terrorist", Trump wrote on Twitter. Ambassador to London Antonio Lagdameo said they are continuously monitoring the situation.
No arrests have been made, and police would not comment on the manhunt.
I utterly condemn the hideous individuals who target innocent people and attempt to use terror to harm us and destroy our way of life.