The Daily Nation reported Saturday it has seen the report, prepared by counter-terrorism agencies. It said the warnings went to the secretaries of the Interior, Defense and Treasury, and Gen. Julius Karang, head of the Kenya Defense Force.
The first alerts came in January and included warnings that al-Shabaab, the Somali group that has taken responsibility for the attack on the Westgate Mall in Nairobi, would attempt to storm a building.
"Briefs were made to them informing them of increasing threat of terrorism and of plans to launch simultaneous attacks in Nairobi and Mombasa around September 13 and 20, 2013," the report said as quoted in the Nation.
Parliament has scheduled a hearing next week on the terrorist attack at Westgate Premier Shopping Mall in Nairobi.
The committee summoned the heads of various security agencies to explain how the raid happened, and how police and the military handled the response.
"The time for responsibility and accountability has come," committee Chairman Ndung'u Gethenji told reporters.
The BBC reported Saturday a sixth Briton had died of injuries suffered in the attack, during which gunmen from al-Shabaab, based in Somalia, went on a shooting spree at the mall a week ago.
The Los Angeles Times reported Saturday there had been reports the government had been tipped off the attack was coming, that the response was both uncoordinated and that the rescue of a trapped group of VIPs had been the top priority of some security units.
The Times said the identity of the supposed VIPs was not known, but the nephew of President Uhuru Kenyatta and his fiancee were among the causalities.
Kenyan media have reported a police commander inside the mall was killed by friendly fire from an army unit that had entered the building. The police withdrew, which media reports said allowed the terrorists to go back on the offensive against shoppers still trapped in the building.
Investigators were at the mall Saturday looking for forensic evidence and any other victims buried in the rubble of a collapsed section of the building.