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A woman called on homes claiming she needed money urgently get back to her son more than 20 miles away.
Now police are treating the two reported incidents as distraction burglaries and are warning residents to be vigilant.
The suspect called on and got inside two flats at Godwyne Close, Dover, between 10.15am and 11am on last Friday Feb 16.
Det Sgt Brian Doughty said: "In both incidents a young woman told residents she needed to get back to Ashford because she’d left her son home alone, and she asked them for money.
"It doesn’t appear that anything was stolen from either property, but one of the victims did give her a small amount of change."
The woman police are seeking described as being about 5ft 7in tall, of a slim or medium build and with an English accent.
She had shoulder-length blonde hair which was tied up in a bun or ponytail, and was wearing a khaki coat or cardigan, a white or pink vest top and denim leggings.
She was carrying a grey handbag.
DS Doughty urged people to be vigilant and to look out for elderly and vulnerable relatives and neighbours.
He said: "Distraction thieves often prey on a person’s good nature, asking for change for the phone or a bus fare or for a glass of water.
"Don't be afraid to turn people away."
Police say people should protect themselves by checking the identity of cold callers and using a door chain or spy hole to do this without letting them in.
"But there are ways people can protect themselves by checking the identity of cold callers and doing so with a door chain or spy hole so they are not let in.
‘Ensure rear doors and windows are shut and locked when answering your front door and consider fitting extra security measures such as spy holes, door chains, door locks and intruder alarms.’
Detective Sergeant Doughty is appealing for anyone who recognises the description of the woman or who has information about the incident.
People should to call Kent Police on 01843 222289 or Kent Crimestoppers anonymously on 0800 555111 quoting reference number ZY/8450/1