The scams were carried out in the Co Down area.
Detective Sergeant Melanie McGrory from Ardmore CID in Newry said: “ In the first incident we were contacted by an elderly couple from County Down to report that they received a telephone call on Thursday from a man, who claimed to work for British Telecom, informing them of a fault on their line.
He asked them to log onto their computer to test various online applications and talked them through a process to allow him remote access to their computer.
“The man has then gained access to their computer and withdrawn £10,000 from their online banking account. In the second incident, again in County Down, a woman was contacted by a man purporting to be from British Telecom, who claimed that he needed her banking details in order to refund her a sum of money.
“After the details were relayed to the man, a sum in excess of £10,000 was taken.
“These examples show us how easily scammers can extract large sums of money from unsuspecting members of the public.
“It is therefore of extreme importance that we all take time to consider who it is we are speaking to when we receive an unsolicited call and not to place our trust in a complete stranger.
“Always be wary of any individual that cold calls you and be especially suspicious of anyone who asks for money, banking or credit card information or who asks to gain access remotely to your computer.
“It is highly unusual for any legitimate company or organisation to demand money over the phone or ask to gain access to your computer.
“If you are at all suspicious about a call that you receive, hang up and phone the organisation that the person is purporting to represent to check their authenticity. Ideally make the call from another telephone so you can be sure the original caller has not remained on the line.
“Guarding your personal and banking details is essential. Never disclose them to any unauthorised person.
“If you have received a call of this kind or are concerned by the intent of unsolicited calls, emails or letters then please report it to Action Fraud via their websitewww.actionfraud.police.uk or by phoning 0300 123 2040, or call police on the non-emergency number 101.
“Further advice and information can also be obtained by visiting www.nidirect.gov.uk/scamwiseni
“And remember, if you can spot a scam you can stop a scam.”