Al-Qaida's claim comes shortly after several people were detained in Saudi Arabia on Tuesday for suspected involvement in the blast.
According to a Saudi weekly published in London, al-Majalla, "al-Qaida has claimed responsibility for the bomb attacks on al-Muhaya in Riyadh this past Saturday. It said in an e-mail message received by one of our correspondents in Dubai that the next strikes will be in the Gulf, America and Iraq".
The message was sent by an al-Qaida member, Abu Muhammad al-Ablaj, who is in regular contact with the publication.
Al-Ablaj warned those "who work and live with Americans", adding that "their killing was permitted" according to religious edicts.
Meanwhile, a diplomatic source close to the Saudi investigation said several suspects were arrested in the capital Riyadh and its outskirts.
An injured child and resident in
hospital after the Riyadh attack
"A group of suspects in the bombing was detained. The campaign to pursue the culprits started immediately after the bombing. Some were also detained and released after interrogation," the source said.
"The campaign to hunt down those responsible is continuing," he added.
Usama bin Ladin's al-Qaida was from the beginning suspected to be behind the bombing in a housing complex, home to mostly Arab expatriates, which killed 18 people and left 120 wounded.
King vows retaliation
Saudi Arabia's King Fahd vowed soon after the bombing those responsible would be dealt with by an "iron fist".
"The retaliation would be stiff," the king told a weekly cabinet meeting on Monday.
The Saudi authorities, meanwhile, had beefed up security in the holy city of Makka, where 2.5 million Muslims were expected to observe the last 10 days of the fasting month of Ramadan.
Security sources said at least 5000 extra soldiers and policemen had been deployed to prevent any untoward incidents.
Security is also being tightened in Madina, the second holiest site in Islam.