The arrests came after a man, posing as a maintenance engineer, allegedly tried to install a keyboard video mouse to a computer at a branch in London, The Telegraph reported.
The user of the device would have then been able to control a network of computers from one machine.
The technology could have allowed the men to steal thousands of dollars from the bank.
Police said they have detained 12 men, between the ages of 23 and 50, on suspicion of conspiracy to steal.
The arrests came after a lengthy investigation, police said.
"This was a sophisticated plot that could have led to the loss of a very large amount of money from the bank, and is the most significant case of this kind that we have come across," Detective Inspector Mark Raymond said.