Police check passers-by near the Champs Elysees in Paris after a shooting which left one officer dead and two wounded.
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.
After killing an officer, he attempted to run away while shooting at other officers, two of whom he injured, the spokesman added.
Another man suspected of possible links to the attack has turned himself in to Belgian police.
France began picking itself up Friday from another deadly shooting claimed by the Islamic State group, with President Francois Hollande convening the government's security council and his would-be successors in the presidential election campaign treading carefully before voting this weekend.
France has been in a state of emergency since a spate of terror attacks hit the country in 2015.
In Washington, President Trump said the shooting in Paris "looks like another terrorist attack".
France's anti-terrorism prosecutor says that searches are under way and three police officials say at least one location in the eastern Paris suburbs was being searched early Friday.
Prime Minister Bernard Cazeneuve said security forces, including elite units, were fully mobilised ahead of Sunday's presidential elections.
As of Friday morning local time, leading candidates François Fillon, Emmanuel Macron and Marine Le Pen had all canceled campaign events, with Macron citing the extra burden policing political rallies placed on the security services.
The Islamic State group claimed responsibility for Thursday's attack unusually quickly in a statement that sowed confusion by apparently misidentifying the gunman.
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An investigation is now underway into the attack, which so-called Islamic State is claiming responsibility for.
British Prime Minister Theresa May also spoke to Hollande.
"Emotion and solidarity for our forces of order, once again targets", Le Pen said after the shooting.
Municipal workers in white hygiene suits were out before dawn to wash down the sidewalk where the assault took place - a scene now depressingly familiar after multiple attacks that have killed more than 230 people in France over two years.
Police sources said the man was known to intelligence services.
"I just want to go home", she said. Two police officers told AP the attacker had been flagged as an extremist, but offered no further details. "As we wait for more definite information, I think we need to attend to our duties as citizens: no panic, we shouldn't interrupt our democratic process", Melenchon said.
13 Jun 2016 -A knife-wielding jihadist kills a police officer and his partner at their home in Magnanville, west of Paris.
The shooter pulled up beside a parked police auto near the Franklin Roosevelt Metro station and opened fire around 9 p.m. local time (3 p.m. ET), he said.
Earlier, Interior Ministry spokesman Pierre-Henry Brandet gave the first official account of the mayhem, saying that the "police officers were deliberately targeted".
Following the shoot out, the Champs-Elysees was closed and authorities were telling people to avoid the area.
3 Feb 2017 - A machete-wielding Egyptian man shouting "Allahu akbar" attacks French soldiers at Paris's Louvre Museum - he is shot and wounded.