Paris police spokeswoman Johanna Primevert said the attacker targeted officers guarding the area near the Franklin Roosevelt metro station at the centre of the avenue popular with tourists.
The suspect, who was shot dead by French police, was the subject of a "Fiche S" surveillance file and was on the radar of the French domestic security service DGSI, the source said.
The man was a French national who shot two officers in 2001 after being stopped by a police vehicle, the source said.
Mr Brandet said the police officers were "deliberately" targeted.
French President Francois Hollande says he is convinced the circumstances surrounding the attack on Paris police officers points to a terrorist act. They believe there was just one attacker, and the danger is likely over, the source said.
Security forces are more widespread in Paris since deadly Islamic extremist attacks in recent years, and France remains under a state of emergency. The ISIS claim comes via a statement released by the group's media wing, Amaq.
France has been plagued by terrorism since the horrific attacks of November 2015 that left 130 people dead and hundreds injured. It was not possible for Al Jazeera to independently confirm the claim.
Prosecutor Francois Molins said that investigators have verified the identity of the gunman in Thursday night's attack and are trying to determine whether he had accomplices. Raids took place at his address in a suburb to the east of Paris.
She said that around the country, even in small villages and towns unaffected by attacks, people still talk about the threat of violence.
"We don't have a precise idea of the origin of the shots fired", he said.
The Champs-Elysees, a top tourist attraction in Paris, is lined with restaurants, cafes, exclusive designer boutiques and tourist shops.
Interior Ministry spokesman Pierre-Henry Brandet said on BFM television that a man came out of a auto and opened fire on a police vehicle. The gunman tried to run, and wounded two other officers as he fled before he was killed.
The gunman was then killed by police. One of the wounded officers was critically injured but is improving, he said.
Video showed the normally busy avenue, one of the most famous streets in the world, empty of residents and tourists but teeming with security officers at 9:30 p.m.
The attack came three days before the first round of France's tense presidential election. Candidates went ahead with a debate Thursday night.
France is now under a state of emergency following a spate of Islamist terror attacks.
Police had arrested two men on Tuesday in what they described as a thwarted terror attack.