Saifullah Khan Ebrahimkhil, the chief of the main hospital in the city, said the casualties included police who it is thought may have been collecting their salaries at the bank.
"In total there are 56 injured and eight dead. Most of the injured and dead are hit by AK-47 rounds," Ebrahimkhil said.
Police are a frequent target for insurgent attacks in Afghanistan as they are due to take responsibility for security with the Afghan army from 2014, by which time most international troops are expected to withdraw.
An AFP reporter at the scene said he had heard gunshots and five explosions as the attack unfolded.
Local officials said that armed men had entered a branch of Kabul Bank and fired weapons. Security forces have now surrounded the area.
The incident was also confirmed by the interior ministry in Kabul.
"An operation is ongoing in the area and we will release further information once the operation is over," interior ministry spokesman Zemarai Bashary said.
Taliban spokesman Zabihullah Mujahed said: "Three suicide bombers have entered the Kabul Bank branch in Jalalabad in the section where they pay the army and police salaries. Big casualties have been inflicted."
Eastern Afghanistan is seen as a particularly volatile part of the war-torn country -- 12 people died in attacks in the region Friday, including nine people in a car bombing near a district police headquarters in the city of Khost.
There are currently around 140,000 international forces in Afghanistan fighting the Taliban, who were ousted by a US-led invasion in 2001 following the September 11 attacks.
The Islamist militants frequently target police and security forces in their battle against the Western-backed Kabul government.