The attacker shot dead by police on the Champs-Elysees on Thursday evening is believed to be Karim Cheurfi, 39.
A property being searched in the eastern Paris suburb of Chelles is understood to be his home and three members of his family have been taken into custody.
Anti-terrorism prosecutor Francois Molins said that a note had apparently fallen out of Cheurfi’s pocket that defended the Islamic State.
Mr Molins said Cheurfi had a long criminal record, including trying to attack police previously.
He added that Cheurfi had been arrested in February but released due to lack of evidence.
French officials, speaking anonymously, said the suspect had been convicted of attempted murder after firing on two plainclothes police in 2001.
While in jail, he also shot and wounded a prison officer after grabbing his gun.
The policeman he killed has been named as 37-year-old Xavier Jugele, by Flag!, a French LGBT police officers’ association.
He was one of the officers who responded to the attack on the Bataclan during the Paris attacks in November 2015.
As the country prepares to vote, France’s prime minister, Bernard Cazeneuve, said more than 50,000 police and gendarmes, as well as 7,000 soldiers, had been mobilised to safeguard the election and protect polling stations.
“Nothing must hamper this democratic moment, essential for our country,” said Mr Cazeneuve.
Far-right candidate Marine Le Pen used the aftermath of the attack to call for the government to immediately expel foreigners being monitored by intelligence services and to reinstate France’s borders.
Conservative candidate Francois Fillon said the fight against “Islamist totalitarianism” should be the priority for France’s next president.
“We are at war, there is no alternative, it’s us or them,” said Mr Fillon.
In a video posted online, frontrunner for the presidency Emmanuel Macron said “the terrorists will is to destabilise the country”.
“In such circumstances, the role of the president of the Republic as the army chief and guardian of our institutions is to protect the French. I am ready,” he said.
The attack happened when a silver Audi pulled up next to a police van on the famous Champs-Elysees avenue – usually packed with tourists heading to the Arc de Triomphe.
The gunman got out and immediately used an assault rifle to shoot officers guarding an area near the Franklin Roosevelt metro station.
He tried to run along the pavement, aiming at other police and hitting two of them.
Other officers opened fire, killing the assailant.
As well as the two officers – who doctors now say are out of danger – a female German tourist was also wounded.
A pump-action shotgun and knives were found in the gunman’s car.
IS claimed it was responsible for the attack and named Abu Yusuf al Beljiki as the attacker.
The pseudonym indicates the attacker was Belgian – something the country’s interior minister, Jan Jambon, has denied, insisting the attacker was French.
He said: “I thought they were firecrackers. In fact, he (the gunman) was hidden behind the van and shooting at the police.
“I think he hit a policeman. As soon as the policeman opened the door of the van, he fell.”
He said he and tourists fled to a shop. “We saw the policeman shoot the gunman who could have killed more of us.”
Police were seen pointing their weapons at members of public as they evacuated the area.
Matthias Fekl, the French interior minister, paid tribute to the dead policeman and praised his colleagues who he said had “prevented a bloodbath”.
US President Donald Trump also tweeted his response to the attack, posting: “Another terrorist attack in Paris. The people of France will not take much more of this. Will have a big effect on presidential election!”