The statement implies that he may have acted alone and there is no evidence that he planned the attacks with them.
The claim of responsibility, reported by the jihadist-linked Amaq news agency, comes as five people were detained following the attack in the French resort city.
However it is not clear if Islamic State are simply seeking publicity from the attack or whether they are claiming responsibility for an attack they may have directed.
Mohamed Lahouaiej-Bouhlel killed 84 people and left scores more critically injured yesterday when he drove a truck through a crowd of people gathered to celebrate France's national day.
The attack killed 84 people and wounded 200.
Police told the Independent that a man actually jumped into the cab and wrestled with the driver. "He would become angry and he shouted... he would break anything he saw in front of him", Mohamed Mondher Lahouaiej-Bouhlel said outside his home in Msaken, eastern Tunisia.
The father also insisted that his son "had no links to religion".
VALERY HACHE AFP Getty Images
Police officers firefighters and rescue workers are seen at the site of the attack
French Justice Minister Jean-Jacques Urvoas said Bouhlel has a criminal record that includes a conviction related to throwing a wood pallet at a motorist.
The new arrests concerned the attacker's "close entourage" and were made in two different areas of Nice.
Another neighbour added that Bouhlel's estranged wife - who had suffered domestic violence at the hands of her husband and was questioned by police on Friday - was a "really lovely woman, who doesn't deserve all this".
Bouhlel, the man accused of seriously injuring an additional 52 people by driving the lorry, was shot dead by armed officers after trading gunfire with police using a 7.65 calibre pistol as he drove.
The attack is the third of its kind in France since the beginning of 2015, and a state of emergency in place since 130 people were killed in and around Paris last November is to be extended for another three months.
The killer also reportedly parked his vehicle on the beachfront before 15:00 (local time) (14:00 GMT) when the roads leading to the promenade were shut as the city was preparing for the celebrations.
The president said that the fight against ISIS and other terrorist groups will require the U.S.to not give up on diplomacy to end the Syrian civil war, for example, and to work with Muslims around the world.