Cashpoints in George Street, Raeburn Place and Boswall Parkway were all targeted between Sunday and Monday by having a plastic covering placed over the dispenser which prevents money from being withdrawn.
The covering is intended to trap the money in the machine before it becomes visible, leading the user to believe that the transaction has failed.
In reality, the cash can be recovered by simply removing the covering, which is disguised to look as if it is part of the ATM.
A police spokesman said: “We are appealing for any information that can help us trace those responsible for placing these devices on cashpoints around the city, and we would urge members of the public who notice anyone tampering with ATMs to contact police immediately.
“In the meantime, we would urge people to study closely any ATM before they use it, and if they notice anything suspicious, to contact police.”