The bank is in the same building as Italy's highest appeal tribunal and whoever planted the container of explosive alcohol would have had to pass through the court's security checkpoints.
Rudimentary bombs made from cans of gasoline were also discovered outside another bank, Banca Toniolo, southeast of the capital, Italian media said. A note attached to them read "Fire to the banks" and "Anarchy".
It was not immediately clear who had planted the devices.
Many Italians believe banks carry the blame for their country's economic crisis because taxpayers were forced to bail out several lenders across Europe.
The country has seen a string of attacks claimed by a group that calls itself the Informal Anarchist Federation (FAI), including a letter bomb that blew off the finger of a tax agency director.