"Our city has suffered two bad terrorist attacks since I was elected as mayor - and we must do everything possible to stop there being any more", said Mr Khan, who praised emergency services after seven people were killed in Saturday night's London Bridge attack.
All three attackers were confronted and shot dead by armed officers within eight minutes of being called, police noted. Family members said the 27-year-old, who once tried to set up a cosmetics company, had also tried to go to Syria to join jihadists but had been talked out of it by relatives.
But the premier also faced mounting criticism for her record on security in the six years she served as Britain's interior minister before becoming prime minister last year.
An off-duty police officer who was stabbed after tackling one of the attackers during the London Bridge attack is in critical condition, the BBC said.
Separately, Butt is claimed to have been reported to police at least twice for expressing extremist views and trying to radicalise teenagers close to his home in Barking, east London.
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The conservative Daily Telegraph commented: 'While we pride ourselves on being an open and tolerant society, it is astonishing that people who pose such a danger to life and limb should be able to parade their foul ideology on TV with no consequences'.
While on Monday a family feared for their son whose bank card was found on the body of one of the victims, however the police have been unable to formally identify the man.
Located in the multi-ethnic area of Barking, the flat where British media reported the man lived was raided by police on Sunday and an AFP photographer saw women in headscarves being taken out - their faces covered.
One of Butt's neighbours, Ikenna Chigbo, told Reuters he had chatted with Butt - known locally as "Abz" - just hours before the attack on Saturday and said he appeared "almost euphoric".
May's chief rival in Thursday election, Labour leader Jeremy Corbyn, has seized on this fact and called for her resignation on Monday.
On May 22, Salman Abedi, a British-born suicide bomber, killed 22 people and injured scores of others at an Ariana Grande concert in Manchester in northwestern England. The bridge had been closed since police shut it down in the midst of the attack on Saturday.
A separate vehicle-ramming attack in March on Westminster Bridge left four dead, including the suspect.
From the United States to the United Kingdom, there were also tweets naming donations made in her honour. London police killed three attackers.