LONDON — ISIS propaganda had been found in the bag of one attacker while he was trying to board a flight in Italy.
An F.B.I. informant said he had raised alarms about the second attacker two years ago. The third attacker, denied asylum in Britain, appeared to have sneaked in from Ireland.
The warning signs about the three assailants in a white van who smashed and stabbed their way through a trendy London neighborhood tumbled into the open on Tuesday, compounding the pressure on the police and Prime Minister Theresa May to explain them.
What has become clear since the Saturday night assault is that again and again, the young men who killed seven people before they were shot to death by the police had been reported to law enforcement authorities, bumping into what should have been the country’s security net, only for those signals to be played down, ignored or missed.
The latest revelations have placed Mrs. May, a former home secretary who was in charge of counterterrorism for six years before taking over as prime minister last year, under intense scrutiny two days before a general election. Even her own foreign secretary, Boris Johnson, a former London mayor, voiced the question many here are asking.
“How on earth could we have let this guy or possibly more through the net — what happened?” he asked in an interview on Sky News.
Some of the missed warnings were especially glaring because they came from the very people the British government had entrusted with identifying extremists.
Usama Hasan, a former Islamic extremist who now works with the police to help de-radicalize others, said he had a physical altercation in a London park less than a year ago with one of the assailants, Khuram Shazad Butt.
Mr. Butt’s brother, Saad, who did paid work for the police on counterextremism issues and was estranged from the assailant, missed signs of how dangerous his brother’s extremism had become.
Other warnings had also been raised about Mr. Butt, 27, who held odd jobs, including at KFC and a six-month stint as a customer service trainee for the London subway system that ended in October. His second child was born weeks before the attack, neighbors said.
In 2015, an F.B.I. informant, Jesse Morton, wrote a report to his handler in the United States, identifying Mr. Butt as a person to watch because of what Mr. Morton described as his rising role in extremist chat rooms run by Al Muhajiroun, an organization banned in Britain because of its sprawling links to terrorism.
“My handler got back to me and said it was ‘excellent work’ and forwarded it to the head office,” said Mr. Morton, a former Qaeda recruiter from New York who served prison time on terrorism charges before recanting and agreeing to work undercover for law enforcement.
Mr. Morton, who recently started Parallel Networks, an organization combating extremism, said it was unclear to him whether his F.B.I. report had been forwarded to British officials. A spokesman for the F.B.I., Andrew C. Ames, said the agency had no comment.
Even excluding the F.B.I. report, plenty of alarms were ringing for the British authorities about Mr. Butt.
Neighbors and friends noticed his behavior, including a mother of three who lived in the same apartment building, said she confronted Mr. Butt two years ago after he tried to convert her son to Islam. When she found him in a local park, she recalled, Mr. Butt said he was ready to do “in the name of Allah what needs to be done, including killing my own mother.”
Ms. Gasparri said she had called a police hotline and passed on photographs she had taken of him, but never got a call back.
Meanwhile, in Italy, the authorities allowed the second attacker, Youssef Zaghba, to walk past them last year at an airport security check, even though he was carrying ISIS propaganda. Mr. Zaghba is an Italian of Moroccan descent, was en route to Syria to fight for ISIS when he was stopped in March at the airport in Bologna.
He was traveling on a one-way ticket, and the authorities found ISIS material on one of his electronic devices, said two former European intelligence officials, who spoke on the condition of anonymity.
Mr. Zaghba was arrested and his belongings were confiscated, but after a judge charged with verifying the accusations against him found there were no grounds to hold him, Mr. Amato said.
Still, Mr. Amato said that Mr. Zaghba had been singled out as a “suspicious person” to the British authorities.
“We did everything we could do,” he said. “But there was no proof he was a terrorist.”
Free to move around in Italy, Mr. Zaghba tapped into that country’s little-known Muhajiroun network, said a retired senior European law enforcement official who was keeping tabs on the investigation. While the dates are unclear, the official said the Italy-based network appeared to have introduced Mr. Zaghba to Mr. Butt.
Al Muhajiroun has been described as perhaps Europe’s most effective jihadist recruitment machine. An estimated one-third of those who have joined ISIS in Syria in recent years were influenced by the network of groups spawned by Al Muhajiroun and one of its founders, Anjem Choudary, a lawyer turned radical preacher.
In Britain alone, at least half of all terrorism cases have publicly documented links to Mr. Choudary, including the 2005 London transit bombings and the killing of a British soldier, Lee Rigby.