Major General Qassim Atta, a spokesman for the security forces in Baghdad, said soldiers and police were "besieging" the attackers whom he described as "a terrorist group."
He said it was unclear if they had intended to rob the bank, target its employees and take hostages, or destroy the building.
The bomb blasts came one day before the opening of the second parliament since the US-led invasion that ousted Saddam Hussein in 2003.
The opening session of the Council of Representatives marks one of the few tangible forward steps taken by Iraq's politicians since a general election on March 7 resulted in deadlock between rival parties.
Diplomats and politicians, warned ahead of Monday's opening that a new government continues to appear some way off, and that it may be several months before the fine detail on the country's new leaders takes shape.
The Iraqiya bloc, led by former prime minister Iyad Allawi, won most seats, 91, in the election, followed closely by prime minister Nouri al-Maliki's State of Law Alliance, which won 89, but both have failed to build a coalition government.
In a sign that the political tempo may be speeding up, Allawi and Maliki held a long-awaited meeting on Saturday, which was described as "friendly and positive," according to a brief statement released by the prime minister.