Police have increased patrols and warned parents after four separate reports of children being followed over the past week.
The Metropolitan Police said two boys were approached separately by a man in the south-east London borough of Bromley on April 30 between 4pm and 4.30pm.
One of the boys, aged 11, was asked by a man in a black van on Whitmore Road in Beckenham if he wanted a lift, while an eight-year-old was approached in the nearby wooded area of Kelsey Park.
The 11-year-old is the son of Line of Duty star Sherise Blackman, who told how the youngster "sensed danger" when he was approached and was left "shaken up".
The Met received another report on Thursday of a male school pupil being followed by a man along a street on the previous day.
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A “lack of similarities” with the incident indicated it was not linked with the first two reports, the force said.
The latest incident involved the “perceived” attempted abduction of a young girl at The Glades shopping centre on Friday.
Two males were said to have been walking behind the girl in the south-east London borough when one attempted to grab her hand as it was raised in the air, before walking off.
The Met said the incident is not believed to be linked to any of the others.
None of the children were abducted or harmed in any of the four reported incidents, the Met said.
Police have written to parents via schools to make them aware of the incidents and encourage them to talk to their children about "personal safety" and remind them to report anything suspicious "immediately" to a parent, guardian or school staff.
A letter stated: "I understand the real concern and anxiety that the nature of these incidents must bring to you and your family; please be assured that we are working closely your schools to provide additional support and advice."
The force is reviewing CCTV and has asked potential witnesses to get in touch.
Superintendent Andy Brittain said: “I am aware of varying reports online both in the news and on social media and I fully understand the concerns of parents in light of these reports.
“I would encourage the public to remain vigilant, but not to be unduly alarmed.
“Child kidnappings or abductions are, thankfully, incredibly rare, but we are not complacent.
“In order to provide reassurance to the community we have heightened our visible presence in areas where youngsters may gather and I urge anyone who feels worried or concerned to approach my officers.
“I would strongly encourage members of the public to approach our officers if they have any concerns or have any information that could prove to be beneficial to our investigations.”
The force has apologised to Ms Blackman, who played officer Ruby Jones in BBC series Line of Duty, after she complained that officers had tried to “fob off” her son's version of events.
Ms Blackman said in a post on Instagram that her son was followed by “hooded men” on foot and another in a car in Beckenham.
She said her son was asked if he wanted a lift and left “shaken up” by the incident.
He ran towards people in a park for help.
Ms Blackman added: “This is real and very serious, we have reported to the police but please keep your kids close and talk to them about these dangers.”
The mum told the Evening Standard she reported the incident to the police and was told that officers would visit their home two days later, on May 2.
But she was left “very dissatisfied” with the way police handled the report.
She said: “They were asking questions like ‘have you got any issues with other boys’ and ‘do you have beef with other kids’
“I could see where they were going with this and I was like no it wasn’t that. My son is a very smart little boy - he sensed danger. But they were trying to fob it off.”
Supt Brittain said: “I fully acknowledge our initial response was not sufficient given the extremely worrying circumstances of this incident.
“The matter was reviewed and detectives are carrying out further investigation which remains ongoing.”
Anyone with information should contact police on 101 or tweet @MetCC.
In an emergency always dial 999.
