Youssef Zaghba is said to have been stopped by Italian officers trying to travel to Syria previous year. Italian officials said suspicions about him were shared with British authorities and his name was subsequently listed in the European-wide intelligence-sharing system.
In Saturday's attack, three men, wearing fake suicide vests, mowed down pedestrians on London Bridge in a van, before slashing and stabbing revellers in Borough Market, a bustling district of late-night bars and restaurants.
His girlfriend was struck and seriously injured by the van.
A second Italian official familiar with the investigation said there had been "an absolute interchange of investigative information with British authorities" in Zaghba's case.
Police on Tuesday also arrested a 27-year-old man in Barking on suspicion of violating the Terrorism Act and a residence there was being searched.
"A very high priority for us is to try to understand whether they were working with anybody else", she told BBC television. "There are three deaths and eight wounded on the French side", Macron was quoted by Fox News as saying.
Butt was investigated by officers in 2015 but they found no evidence he was planning an attack and he was "prioritised in the lower echelons of our investigative work", police said.
Her opponents say cuts she spearheaded as home secretary have left Britain more vulnerable to terrorism, after three attacks in less than three months.
Italy has expelled more than 40 people in the past two years who were suspected of extremist activities but for whom there was insufficient evidence to bring formal charges.
In a statement, her family said: "We are deeply saddened at the tragic loss of our lovely daughter and sister of Harrison and Scott". Meanwhile, days after 130 mosques refused to bury the Manchester Arena bomber, the Muslim Council of Britain issued a statement condemning the London attack.
A man arrested in Limerick in connection with the London terror attacks was released without charge yesterday afternoon.