In the latest incident, a man was arrested on Saturday morning (April 13) near the Bank of Scotland at Bearsden Cross after a chase by officers on foot, assisted by members of the public.
He was charged with attempted theft and was due to appear at court this week.
Officers praised the vigilance of local people with regard to the arrest and said they are also looking into a number of incidents of scammers using “skimming” devices to take cash from people’s bank accounts.
Card skimmers can be clipped on to the card readers of ATMs and bank machines to silently steal your details .
Crooks can then use this stolen data to make fraudulent transactions online, or sell your data on to give other criminals access to your account and take money.
Sometimes a hidden camera will be placed near a card skimmer to capture your pin number as well, making it easier for the criminals to get into your account.
A police spokesperson said: “We were alerted about Saturday’s incident. It was good to see members of the public looking out for each other.
He added: “There have been other reports of similarincidents. The prevention banks now have is immense but I would urge people to be extra vigilant over these devices.
“If you are in any doubt, contact the police or the bank.”
On social media site, local people have also been warning others about scams linked to cash machines.
One local woman posted on a community Facebook site on April 6: “The machine took our card and we had £400 taken from our account by scammers.”
Later, she added: “The fraud team have been fab and we got the money back - £500 in total.”
Another posted: “I try to not use machines outside branches.
“I go inside the bank and assume it’s safer because of all the staff who are around.”