Abedi’s brother Hashim, who was arrested in Tripoli in the wake of last month’s attack, knew the 22-year-old was planning something, counter-terror official Ahmed bin Salem said.
Hashim, aged 20, said Abedi became radicalised while living in Manchester in 2015, Mr bin Salem told the Associated Press.
The British-born extremist of Libyan descent killed 22 people when he detonated the device as concert-goers began leaving a show by US singer Ariana Grande in Manchester Arena.
Abedi’s father Ramadan was also arrested in Libya two days after the attack.
Both Hashim and Ramadan are still being held for questioning.
On Wednesday, Greater Manchester Police said a 20-year-old man was arrested on suspicion of terror offences in connection with the bombing after handing himself in to police.
To date, 21 people have been arrested in connection with the investigation, of whom 12 have been released without charge and nine remain in custody.
Two of the other arrests came on Tuesday, with a 20-year-old being held in Harpurhey, and a 38-year-old being detained at Heathrow Airport as part of a planned operation, both on suspicion of terrorism offences.
Also on Tuesday, police said they had found “significant evidence” in a car where Abedi may have stored items used to assemble his improvised explosive device.
It is reported Hashim also told Libyan officials Abedi bought materials with which to carry out the atrocity in Britain.
The inquests into the deaths of the victims, who include seven children, are due to begin on Friday at 10.30am.
They will be opened at Manchester Civil Justice Centre by senior coroner for Manchester Nigel Meadows.
The hearings are expected to be adjourned, pending the completion of the police investigation.
An inquest into Abedi’s death will be opened at a later date.
On Tuesday, an 18-year-old man, understood to be Abedi’s cousin, and two other men were freed.
Abedi’s older brother, Ismail, 23, was released on Monday, while his other cousins, Isaac Forjani, 24, and Abz Forjani, 21, both from Fallowfield, were among those freed last week.
Islamic State claimed responsibility for the atrocity.