According to Italy's Corriere della Sera newspaper Zaghba had been stopped at Bologna airport in 2016 when trying to fly to Syria via Turkey, and Italian authorities had identified him as as a potential "foreign fighter" and tipped off Britain about his movements.
Butt had been a key player in terrorist group al-Muhajiroun since at least 2009 and had been mentioned in an Federal Bureau of Investigation report in 2015, the paper reported.
He was prevented from continuing his journey to Istanbul, placed on a watch list and flagged to Moroccan and British counterparts, it was claimed. In particular, there are questions over whether investigators had the resources to look into complaints such as those leveled by Butt's neighbors about his attempts to radicalize children and whether crucial opportunities were missed.
Xavier Thomas, 45, is thought to have been struck by the terrorists' van on the bridge and witnesses reported him being thrown into the water.
London police have said all 12 people from the Barking neighborhood held since the attack have been freed.
Mark Rowley, the Metropolitan police assistant commissioner, said the changes could cover police, MI5, communities, technology companies, the law and other policies. Whatever happened, there are now serious questions for the authorities to answer about what went so badly wrong.
Seven people were killed and 48 injured in the attack on Saturday night.
Police said three men - Khuram Shazad Butt, 27; Rachid Redouane, 30; and Youssef Zaghba, 22 - rammed a van into pedestrians on London Bridge before launching a stabbing spree in bars and restaurants at nearby Borough Market on Saturday.
Security source says Khuram Butt was well known to police.
"I feel like I did what any other person would have done", he said.
Armed police responded and within eight minutes had shot dead all three attackers.
One of the victims of Saturday's terror attack has been confirmed as Australian Sara Zelenak.
Police said Zaghba had not been a subject of interest to police or the domestic spy agency MI5.