Raids took place at the 39-year-old's Paris suburb home during the night after he killed the policeman and wounded two others in an attack claimed by the Islamic State group.
However, the French government is determined that Paris will not be closed down by these continued terrorist incidents, with the Champs-Elysees boulevard now reported to have been reopened - albeit under tight security controls.
Investigators say they found a pump-action shotgun and knives in Cheurfi's abandoned auto.
It was unclear how the election would be impacted by the shooting, which came days after two men were arrested in Marseille on suspicion of plotting an imminent attack.
Mr Hollande held an emergency meeting with the prime minister on Thursday night and will convene a meeting of the defence council on Friday morning.
So-called Islamic State (IS) said one of its "fighters" had carried out the attack.
The official and another, also speaking on condition of anonymity, said Cheurfi was detained toward the end of February after speaking threateningly about police but was then released for lack of evidence.
The offices of Charlie Hebdo magazine were hit in January 2015, IS gunmen and suicide bombers killed 130 people at sites around Paris in November that year, and a Tunisian man rammed a truck through crowds in Nice in July 2016, killing 86 people.
The extremist group gave a pseudonym for the gunman, Abu Yusuf al-Beljiki, indicating he was Belgian or had lived in Belgium, although Belgian authorities said they had no information about the suspect.
Authorities are trying to determine whether "one or more people" might have helped the attacker, Brandet said.
NPR's Eleanor Beardsley reports from the Champs Elysees that the avenue has been completely cordoned off. "People are not allowed on it anymore".
He had also wounded two police officers. She cancelled a minor campaign stop, but scheduled another.
Spokesperson of the French interior ministry Pierre-Henry Brandet told local broadcaster BFMTV that the police officers were "deliberately targeted", and the assailant was shot dead.
REUTERS/Christian Hartmann Forensic experts collect evidences from the vehicle belonging to an attacker who killed a police officer on the Champs Elysees avenue in Paris, France, Friday, April 21, 2017.
Meanwhile, following the attack, the presidential candidates have called off their election campaign.
Anti-terror prosecutors immediately took up the investigation and President Francois Hollande said it was "of a terrorist nature".
"And we need to strengthen and improve coordination and that is why one the pledges I have made if I were elected, is to create within the first few weeks an anti-Daesh Task Force under the president's direct authority, which will be in charge of coordinating all intelligence services".
Security in Paris was increased this week ahead of the French presidential election on Sunday.
German Chancellor Angela Merkel and U.S. President Donald Trump expressed their condolences to French President, with Trump labeling the shooting as a "terror attack".