Police have not yet formally released the suspect's name, but the Gazette's sister paper in Weston understands he is 47-year-old Darren Osborne, who grew up there.
Police have confirmed they are carrying out a search at "residential address in the Cardiff area".
Officials said 11 people were injured, including nine taken to hospital and two treated at the scene.
As of mid-Monday morning, U.S. President Donald Trump has yet to respond, via his prolific Twitter account or by any other means, since a man slammed his vehicle into a crowd of Muslim worshipers just after midnight.
"As a community it is important to show that we stand together and that we will not be bowed by this attack or the others we have seen in Manchester and London in recent weeks".
"Hatred and evil of this kind will never succeed", May said.More news: Expulsion of Qataris from Gulf states comes into effect
London Mayor Sadiq Khan said there would be an increased police presence around mosques in coming days to protect Muslims from hate crimes.
One man, who was already ill and receiving CPR on the side of the road as the attack occurred, passed away after the incident.
Witnesses said the attacker was shouting that he wanted to "kill all Muslims". "One eyewitness told the BBC that he heard the driver saying "'I want to kill Muslims.
After being dragged from the van by an angry mob, he was protected by the Imam of the mosque, Mohammed Mahmoud, who ordered people not to attack him, but hand him over to the police. "We stopped him until the police came".
Metropolitan Police Commissioner Cressida Dick added: "We are here as you are well aware because of the awful, bad attack that took place last night".
Labour leader Mr Corbyn, who is also the local MP for Islington North, was greeted with a warmer reception as he condemned the incident as "an attack on all of us".
On March 22, a man drove a rented auto into pedestrians on Westminster Bridge in London and stabbed a policeman to death before being shot dead.