Both Butt and Zaghba were known to British law enforcement, raising questions about whether anything could have been done to prevent the assault.
Scotland Yard said formal identification had not yet taken place, but Mr Thomas' next of kin had been informed.
Butt was known to police and "openly fundamentalist", NPR's Eleanor Beardsley reports from London.
Although news of her son's death came as a shock, she said she could only share the pain of the victims' families.
Butt is a 27-year-old British citizen who was born in Pakistan. He and Butt lived in Barking in east London.
The second attacker was named as 30-year-old Rachid Redouane, who police said claimed to be Moroccan and Libyan, and also went by the name Rachid Elkhdar with a different date of birth. All three attacks have been claimed by the Islamic State group.
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A 27-year-old was arrested on suspicion of the preparation of terrorist acts and a 33-year-old was arrested on suspicion of possession with intent to supply controlled drugs, police said.
A minute's silence is due to be observed in Britain at 11 a.m. local time (1000 GMT) in memory of the seven people who died during the attack late Saturday. The men were followers of Anjem Choudary, a preacher who was jailed for his support of the Islamic State and who once praised the September 11 attackers. Authorities said he aroused suspicion because he tried to board a flight to Turkey with a one-way ticket and a small backpack.
Police searched his home but no "particular elements" were found apart from religious documents that were downloaded from extremist sites, ANSA reported.
Four of those killed in the attack have been named as Australian Kirsty Boden, Canadian national Chrissy Archibald, James McMullan, from Hackney, and French national Alexandre Pigeard.
But an announcement - made before the Manchester and London Bridge attacks - that they planned to make some of the elderly pay more for their care saw that lead start to shrink, and the trend has continued.
The officials spoke on condition of anonymity because they were not allowed to discuss details of the case. The three attackers were shot dead at the scene by police.
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"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?'"
Earlier police admitted that one of the other attackers was on their radar as a member of an outlawed radical Islamist group, but said an investigation into him had been downgraded as there was no evidence he posed an imminent threat.
Police said Zaghba is believed to be an Italian national of Moroccan dissent and was not a "police or MI5 subject of interest".
He and two other suspects mowed down numerous people on London Bridge Saturday night, before running through the streets to stab several more.
Police said earlier that witness accounts suggested he might have been thrown into the river.
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