One of the plotters posed as an engineer to fit a computer in a branch of Santander with a "keyboard video mouse" device that allowed them to transmit its desktop contents.
Using the gadget, the gang remotely took control of all the computers at the branch in Surrey Quays shopping centre in London, but the Spanish bank said they were unable to steal any money.
Officers have arrested 11 men aged between 23 and 50 in Hounslow, west London. A 34-year-old was arrested in Victoria, west London.
Searches were carried out at addresses in Westminster, Hounslow, Hillingdon, Brent and Richmond, and also in Slough, where property has been seized.
Detective Inspector Mark Raymond, from Scotland Yard's Police Central e-crime Unit (PCeU), 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.
"I would like to thank our partners from the industry who have provided valuable assistance throughout this investigation."
Santander said that no member of staff was involved in the attempted raid.
A spokesman said: "Like all the high street banks, Santander works very closely with the police and other authorities to help prevent fraud.
"Through this co-operation, Santander was aware of the possibility of an attack. It failed and no money was ever at risk."