Government officials said almost 1,000 soldiers were deployed to Buckingham Palace, Parliament and other high-profile sites across the country.
British media indicated that this man is highly likely to be Abedi's older brother Ismael.
Prosecutors said Abdallah, who denied links with ISIS, he was "at the centre of a jihadist network facilitating foreign fighters".
Also, officials were looking into reports that people who knew Abedi - including an imam at his mosque - had contacted authorities as early as two years ago with concerns that he had become radicalized.
British Prime Minister Theresa May, arriving at a North Atlantic Treaty Organisation summit in Brussels, said the countries' partnership on defence and security was built on trust, but "part of that trust is knowing that intelligence can be shared confidently".
The Telegraph newspaper reported that Abedi was born in Manchester and was one of four children to parents who were Libyan refugees who fled to Britain at the height of the Gaddafi regime. The authorities have raised the threat level in the nation from "severe" to a "critical" level.
In the Libyan capital Tripoli, Abedi's younger brother 20-year-old Hashem and their father, Ramadan, are being held by special forces linked to the country's interior ministry.
A French official has confirmed that Abedi traveled to Syria.
Hashim Abedi's father, Ramadan Abedi, was detained in Tripoli on Wednesday. The Palace of Westminster, which houses Parliament, was shuttered to those without passes, and tours and events were cancelled until further notice.
A former unnamed British intelligence chief toldNewsweek that United States sources were unreliable with sensitive information, claiming they "just blabbermouth it and it's really, really unhelpful".
Neighbors recalled him as tall, thin and quiet, and said he often wore traditional Islamic dress.
Security sources said: "Counter-terrorism policing has temporarily ceased sharing information with U.S. law enforcement partners relating to the Manchester investigation only".
As the search for the suspected bombmaker continued at pace today, MI5 and police faced claims that they were repeatedly warned about Abedi, including allegedly by a member of his family.