Medical officials said many of the victims were Iranian pilgrims, while Iranian state television said there were five Iranian dead and 22 wounded.
Witnesses said a yellow minibus exploded next to two tour buses filled with Iranian pilgrims that had pulled into the parking lot of the Maytham al-Tammar shrine, which honours one of Imam Ali's companions and is a staple on the Shiite pilgrimage circuit.
"The Iranian buses arrived and parked in front of the main gate of the shrine and just as the pilgrims were getting out, the bomb went off," said a local shopkeeper.
"Most of my group of pilgrims was already inside the shrine when the bomb exploded," said an Iranian pilgrim.
The shrine is next to the grand mosque of Kufa, built on the site where Ali, the first imam of the Shiite sect, was assassinated in 661 AD, and is regularly visited by Iranian and Afghan Shiites, in addition to Iraqi Shiites.
Together with the nearby holy cities of Najaf and Karbala, the shrines of Kufa represent some of the world's holiest sites for Shiites.
The Kufa mosque and the Maytham shrine were used by firebrand Shiite cleric Moqtada al-Sadr's Mehdi Army in a 2004 bloody rebellion against US and Iraqi forces to stock weapons, plan attacks and recruit fighters.
Sadr often delivers his Friday sermons from the mosque.