Police have increased their presence in Bromley after receiving 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 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 Met received another report on Thursday of a male school pupil being followed by a man along a street on the previous day.
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 Glade 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.
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.”
None of the children were abducted or harmed in any of the four reported incidents, the Met said.
Anyone with information should contact police on 101 or tweet @MetCC. In an emergency always dial 999.