Zaghba was stopped at the Bologna airport by Italian airport police in March 2016 as he was trying to board a plane bound for Istanbul. Police found what they described as extremist material on his cellphone.
They weren't able to arrest Zaghba, who was traveling on an Italian passport, because having extremist material on a phone isn't against the law in Italy.
Italian media reported that officials suspected he was bound for Syria and that Italian authorities had alerted their British counterparts about his movements.
Police say Zaghba, 22, was one of three men who rammed a van into pedestrians on London Bridge before launching a stabbing spree in bars and restaurants at nearby Borough Market on Saturday night. At least seven people were killed and dozens were injured. The three terrorists were shot dead by police.
Zaghba's reported Syria travel plans raise further questions about whether British authorities could have done more to prevent the attack.
London's Metropolitan Police Service said Zaghba was not a "person of interest" before the attack.
Earlier, London police admitted that one of the other two attackers, Khuram Shazad Butt
, was on their radar as a member of the outlawed radical Islamist group al-Muhajiroun, co-founded by notorious hate preacher Anjem Choudary. British police and MI5 began investigating Butt intensively in 2015
as part of a major drive to dismantle and destabilize al-Muhajiroun, the sources told CNN.
Butt also appeared in a 2016 television documentary called "The Jihadis Next Door."
Assistant Commissioner Mark Rowley said an investigation into Butt had been downgraded as there was no evidence he posed an imminent threat.
The other member of the trio, Rachid Redouane, a 30-year-old who had claimed to be Moroccan and Libyan, was not known to intelligence services.
Irish police arrest man after London Bridge attack
Police in Ireland on Tuesday arrested a man as they pursued a strand of the investigation into the weekend attack, they told CNN. Authorities said the man was not arrested in direct connection with the attack, and was being held at Wexford police station in the southeast of Ireland.
Irish police arrested and released another man in Limerick on Monday.
Appeal for help finding potential victim
On Tuesday, London police appealed for the public's help in finding French national Xavier Thomas, 45, who has been missing since Saturday's attack.
Metropolitan Police said in a statement that Thomas was in town for the weekend with his girlfriend.
According to witnesses, it's possible Thomas -- whose girlfriend was seriously injured -- was struck by the van and thrown into the River Thames. Police ask any other witnesses who were on the bridge or who have seen Thomas to contact them.
Attacker slips through the net
Foreign Secretary Boris Johnson said that the country's intelligence agencies had questions to answer. He told the BBC: "People are going to look at the front pages today and they're going to say, 'How on earth could we have let this guy or possibly more through the net?'"
Just two days before the UK election, Prime Minister Theresa May has faced a barrage of criticism for cutting 20,000 officers from police forces in her time as home secretary.
A Met Police spokesperson told CNN that an investigation into Butt started in the summer of 2015, during which police received a call to the anti-terror hotline. The probe continued but was moved into "the lower echelons of investigations."
"Looking back over the information we had at the time, so far, there is nothing to show that poor decisions were made. We will probably discover communications during that period that we couldn't access that would have change the situation," the spokesperson said.
Raids in London and Pakistan
Raids continued in east London on Tuesday, and detectives arrested a man at an address in Barking. Twelve others arrested after the attack have been released without charge, police said.
The investigation appears to have reached as far as Pakistan, where law enforcement agencies raided a restaurant and home overnight belonging to a man they believe is a relative of Butt. Butt was a 27-year-old British citizen born in Pakistan.
An intelligence official with direct knowledge of the situation said that the relative is an influential businessman in the Jhelum district of Punjab province. "Five or six big trucks were parked outside the restaurant late last night when the raid took place. [His] house was also raided and family members questioned," the source said.
Officials in plain clothes searched the restaurant and questioned people who were there, the source said, adding that no arrests were made.
Butt is not believed to be linked to any militant groups in Waziristan, a known hotbed for militant networks in Pakistan, according to the intelligence source.
Butt, who also used the name Abdul Zaitun, appeared in a 2016 documentary called "The Jihadis Next Door," which told the story of Abu Rumaysah, the Londoner who has appeared in recent ISIS propaganda. At one point in the documentary, Butt can be seen unfurling a black banner in Regent's Park.
A London mosque said Butt "infrequently" attended prayers there but was asked to leave after an incident a few years ago where he interrupted a Friday sermon, according to a statement from Jabir bin Zayd Islamic Centre.
In two videos viewed by CNN, Butt is among a group of men searched by police in 2015 after a member of the public reported seeing the group praying in a park with what appeared to be an ISIS flag.
The ISIS-linked Amaq News Agency claimed a "detachment of Islamic State fighters" carried out the attack but provided no evidence to back up its claim.
May under fire
Despite a rapid response to Saturday's deadly attack, which saw armed officers shoot dead all three suspects within eight minutes
, opposition parties have called for May's resignation for her cuts to police numbers.
Asked whether he backed calls for May's resignation just days before the country's General Election, main opposition Labour Party leader Jeremy Corbyn told ITV, "indeed I would."
He later clarified: "I think we should vote on Thursday to decide who our MPs are and who are government is going to be ... There's an election on now, there's a choice for everybody. There's a deep anger from those people who've seen 20,000 police officers lose their jobs."
Corbyn's party colleague, London Mayor Sadiq Kahn, told Britain's Channel 4 News that the police force is underfunded.
"We simply don't have the resources we need to provide the best service we can," he said in an interview following Monday's vigil.
Since 2010, when the Conservatives entered government, the number of police officers has fallen from 145,948 to 126,766 in March 2016, according to Home Office figures cited by ITV.