The warning comes as Action Fraud has launched a new national social media campaign.
Courier fraud is when a fraudster contacts the victim over the phone and pretends to be either a police officer or bank official.
One victim aged 86, received a call from someone claiming they were from Hammersmith Police and tried to swindle them out of £4,800.
Lucking, bank staff recognised the signs of fraud, and no loss was incurred.