Ten policemen were among the dead in the attack, according to a police official in the city who spoke on condition of anonymity because he was not authorized to release the information.
Fallujah, in restive Anbar province, is 40 miles west of Baghdad.
Police said the bomber detonated his explosives vest at the third of four checkpoints as he stood among recruits who were lining up to apply for jobs on the force. The center had only been opened on Saturday in a primary school in eastern Fallujah.
The U.S. military and Iraqi army and police were running the center along with members of Anbar Salvation Council, a loose grouping of Sunni tribes that have banded together to fight al Qaeda.
Police stations and recruiting posts have been a favorite target of Sunni insurgents and al Qaeda through the course of the Iraq war.
Meanwhile, as the aggressive search for five kidnapped British citizens continued around Baghdad, a Pentagon spokesman acknowledged that the hunt for two U.S. soldiers missing since an ambush on May 12 has slowed down.
Brigadier General Perry Wiggins says the military is still dedicating a "significant amount of assets" to the search for the two soldiers, but says it's not as intense as it was.
The troops were ambushed south of Baghdad nearly three weeks ago. Four soldiers were killed in the initial attack and the body of a third kidnapped soldier was found days later.
Wiggins says there has been a side benefit to the search. Patrols have uncovered weapons caches and bomb-making facilities.
The general says the hunt is also continuing for the five British citizens kidnapped Tuesday in a daring raid on an office of the Iraqi Finance Ministry. He says the U.S. is doing all it can to help find them.
A procession of mourners, some of them women wailing and beating their chests, marched Thursday through Sadr City behind a small bus carrying the coffins of two people who police said were killed in a U.S. helicopter strike before dawn.
The U.S. military did not immediately respond to a request for comment on the alleged attack in the second day of a search for the Britons who were abducted from a Finance Ministry data processing building in eastern Baghdad.
A U.S. military statement, however, said U.S. and Iraqi forces had arrested two "members of the secret cell terrorist network" on Thursday in Sadr City. There was no mention of fatalities.
Associated Press Television news video tape from Sadr City showed the coffins of the victims atop a small bus with men and women walking behind, crying. A young boy could be seen sitting next to the coffins on the bus.
A car near where the attack happened was punctured with big holes as if hit by an air strike.
A police officer in Sadr City, who refused to allow use of his name because he feared retribution, said the helicopter hit a house and car at 4:30 a.m., killing two elderly people sleeping on the roof of their home, a common practice in the extreme heat of Iraq through late spring and summer.
The officer said a 13-year-old boy was injured.
Also in Sadr City raids, which the U.S. has been conducting with a select unit of Iraqi army forces, Shiite cleric Abdul-Zahra al-Suwaidi claimed his home was raided and ransacked by American forces in the early morning hours Thursday.
Al-Suwaidi, who runs the Sadr City political office of radical cleric Muqtada al-Sadr, said he was sleeping elsewhere at the time of the raid, expecting that he would be targeted. He said his home was badly damaged and a small amount of money was taken.
The U.S. military also did not immediately comment on al-Suwaidi claim.
Dozens of U.S. Humvees and Bradley fighting vehicles had taken up positions around Sadr City at nightfall Wednesday.
In other developments: