Youssef Zaghba is said to have been stopped by Italian officers trying to travel to Syria previous year and added to an global database of alerts about individuals, to which the United Kingdom has access.
The other two attackers were named Monday as Khuram Shazad Butt and Rachid Redouane.
Police have arrested three people on terror charges, but they have said they are not linked to the London Bridge attack.
In a separate investigation not linked to the London Bridge attacks, officers backed up by armed police arrested three men in east London on Thursday on suspicion of preparing for acts of terrorism.
The other two attackers were named earlier as Khuram Shazad Butt and Rachid Redouane.
Butt featured in a Channel 4 documentary a year ago about Islamist extremists with links to the jailed preacher Anjem Choudary called The Jihadis Next Door.
Police said Zaghba had not been a subject of interest for them or for the MI5 domestic intelligence agency.
Following the attack, London Mayor Sadiq Khan said the threat level in the country is still "severe".
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Metropolitan Police distributed this photo of Youssef Zaghba, who they have named as the third London Bridge attacker.
Zaghba - who was born in the Moroccan city of Fez in 1995 - had broken off relations with his father, Reuters reported, citing the Corriere della Sera.
Butt was known to police and was "openly fundamentalist", NPR's Eleanor Beardsley reports from London.
The three terrorists were dead - eight minutes after they drove their van into London Bridge and started stabbing anyone they could find.
Thomas, a French tourist to Britain, was found on Tuesday night in the water near Limehouse, East London by the Metropolitan Police, a spokesman for the police said Wednesday.
The parents lived for a time in Morocco before separating when the mother returned to Italy and re-established herself in Bologna.
When Zaghba entered Britain, staff at passport control should automatically have been alerted by the Schengen system, BBC home affairs correspondent Danny Shaw said, citing unofficial sources.
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