Baghdad security spokesman Major General Qassim al-Moussawi gave no details and offered no evidence for the claim and it was not possible to verify it.
The detention of the Saudi national comes at a time when already frosty relations between Sunni Muslim power Saudi Arabia and now Shi'ite Muslim-led Iraq are under additional strains.
Moussawi's allegation about a Saudi's involvement in a plot against the World Cup came after former Saudi intelligence chief Prince Turki al-Faisal accused Iraq's prime minister of hijacking the country's March election.
Moussawi, who reports to the prime minister's office, said Abdullah Azzam al-Qahtani was a former Saudi army lieutenant.
"He was planning to bomb the holy shrines in Najaf and Kerbala," Moussawi told a news conference in Baghdad.
"And he was planning a terrorist act in South Africa during the World Cup based on plans issued by the central al Qaeda terrorist organisation in coordination with Osama bin Laden's first assistant, Ayman al-Zawahri."
Calls to Moussawi seeking more details were not answered. U.S. military officials in Iraq had no knowledge of the allegations, nor did U.S. intelligence officials in Washington.
In South Africa, police spokesman Colonel Vishnu Naidoo said the first that police heard about the claims was in media reports out of Iraq.
"The police will investigate the allegations and we will check with the authorities that apprehended the suspect," Naidoo said.
South Africa is the first African country to host the World Cup, which takes place June 11-July 11.
South African officials have said in the past that no particular terrorist threat had been identified but that there could be risks to teams known to be a target of extremists.
Africa's biggest economy has one of the highest rates of murder and rape in the world and 41,000 police will focus on protecting foreign and local fans against criminals in one of the world's most violent countries outside a war zone.
ICEY RELATIONS
Moussawi did not say why it took more than two weeks to announce Qahtani's capture. But it came two days after the Saudi criticism of his boss.
According to reports, Saudi Prince Faisal on Saturday reproached Iraq's Shi'ite Prime Minister Nuri al-Maliki and warned in a speech of spiralling violence as U.S. troops prepare to leave by an end-2011 deadline.
"Adding to the brutal mayhem taking place there, we are watching a deliberate effort on the part of the incumbent Prime Minister, Mr. Nuri al-Maliki, to hijack the results of the elections and deny the Iraqi people their legitimately elected government," he said, according to the online Wall Street Journal.
Relations between Saudi Arabia and Maliki's Shi'ite-led government have been difficult.
Saudi Arabia views Shi'ite Iran's growing influence in Iraq since Sunni dictator Saddam Hussein was ousted in the 2003 U.S.-led invasion with suspicion. For his part, Maliki has often accused unnamed G
