The calls are thought to have been made five years ago after Abedi left school, the community worker added.
It comes after NBC News cited US officials as saying the bomber’s family had warned security services that he was “dangerous”.
Twenty-two people were killed and dozens more seriously injured when Abedi, 22, detonated a device as fans left Manchester Arena, where US star Ariana Grande was performing on Monday night.
It was the deadliest terrorist incident to hit the UK since the July 7 attacks in London in 2005 and sparked a nationwide security operation amid fears further strikes could be imminent.
On Wednesday afternoon, Greater Manchester Police Chief Constable Ian Hopkins said the level of activity in the probe is “intense” and continuing “at pace”.
Asked if officers are looking for the person who made the bomb, he said: “I think it’s very clear that this is a network that we are investigating and as I’ve said, it continues at pace, this extensive investigation is going on and activity taking place across Greater Manchester as we speak.”
Cheshire Police confirmed that one of its female officers died while off-duty at the concert but have not named her.
The BBC reported that her husband is critically ill and her two children injured.
Meanwhile, Ramadan Abedi, who is in Tripoli, Libya, denied his son was linked to militants or the bombing.
Mr Abedi told the Associated Press his family “aren’t the ones who blow up ourselves among innocents”.
He said his son sounded “normal” when he last spoke to him five days ago, adding that Abedi had visited Libya a month-and-a-half ago.
Meanwhile, it was reported that the bomber’s younger brother, Hashem Abedi, had also been detained in Tripoli on suspicion of links with Islamic State.
With Britain on critical alert for further attacks:
• Three men were arrested after police executed warrants in south Manchester overnight, while officers entered an address in the city centre using a controlled explosion on Wednesday afternoon;
• Another arrest was made later in Wigan of a man carrying a suspect package.
• France’s interior minister disclosed that the bomber is believed to have travelled to Syria and claimed he had “proven” links with Islamic State;
• Almost 1,000 military personnel were being deployed around the country, including to key sites such as Parliament, Buckingham Palace and Downing Street, after the official terror threat assessment was raised to critical, the highest level, indicating that a further attack may be imminent;
• Claims emerged in America that members of the bomber’s family warned security officials about him in the past;
• The Government announced that a minute’s silence will be held at 11am on Thursday in remembrance of those who lost their lives or were affected by the attack;
Prime Minister Theresa May will chair a meeting of the Government’s Cobra emergency committee in Whitehall on Thursday morning, said Downing Street.
The meeting, bringing together senior ministers, officials, police and security agencies, will be the fourth Cobra gathering to deal with the Manchester bombing.
Mrs May announced late on Tuesday that the UK’s terror threat level was being raised to the highest category for the first time for nearly a decade.
In response to the heightened threat, the Government has activated Operation Temperer, providing up to 3,800 troops to support the police in their security operations.
Ms Rudd said the move was temporary but refused to be drawn on how long the heightened state of alert would remain in place.
The Home Secretary said 984 military personnel were being deployed around the country.
France’s interior minister Gerard Collomb told French television that both British and French intelligence services had information that the attacker had been in Syria.
Mr Collomb said: “We only know what the British investigators have told us. He was a British national of Libyan origin, he grew up in Great Britain.
“All of a sudden he travelled to Libya and then most likely to Syria, became radicalised and decided to commit this attack.”
Ms Rudd confirmed Abedi had recently returned from a visit to Libya,and said the nature of the attack suggested he may have had support.
“It was more sophisticated than some of the horrific events that we have seen in the past or in other parts of Europe so people are reasonably wondering whether he did this on his own,” she said.
The Islamic State terror group claimed responsibility for the barbaric attack, which involved a home-made device packed with nuts and bolts which exploded in the venue’s foyer as thousands of young people were leaving.