But the statement quoting a security member of the group said the man was following calls from IS to target nationals of countries fighting it.
Mr Hollande had proposed lifting the state of emergency on 26 July, but reversed his decision after the Nice attack.
French authorities are still trying to determine whether Lahouaiej-Bouhlel acted alone or with accomplices, and whether his motives were connected to radical Islam.
The driver's father has said that Bouhlel had received psychiatric treatment in the past.
Cazeneuve said the carnage had "deeply shocked the French and at the same time shows the extreme difficulty of the anti-terrorism fight".
Another neighbour, who did not want to be named, said she knew Bouhlel's wife and described her as a "really lovely woman, who doesn't deserve all this". He was not known to French intelligence sources for radicalization.
A sunflower is seen on a fence in front of the Negresco hotel, on the Promenade Des Anglais, in memory of victims the day after a truck ran into a crowd at high speed killing scores and injuring more who were celebrating the Bastille Day national holiday, in Nice, France July 15, 2016.
Defense and intelligence leaders met with President Francois Holland in Paris on Saturday to discuss the third mass-casualty attack to hit France in the last 18 months. During a speech to the nation on Friday, President Francois Hollande said that in addition to the 84 people killed, 50 remained in critical condition.
Defence Minister Jean-Yves Le Drian said a special terrorism vigilance force created after the 2015 attacks will be extended through the summer, with more deployments outside Paris and more attention to tourist sites and crowded events.
Bouhlel had a criminal record, including a history of threats, violence, petty theft and earlier this year was sentenced to six months in prison for a road rage incident.
While Islamic State (IS) has described the lorry driver who massacred at least 84 people in Nice as "a soldier" who responded to calls to attack its enemies, people who knew him personally have described him as a man with "psychological problems" - and not someone they even considered to be a devout Muslim.
One of the men being held was arrested Friday and three others on Saturday morning, the source added.
Police have arrested four more people linked to Lahouaiej-Bouhlel, as well as his estranged wife. "I don't want this to change how we feel about France", she said as she wiped away a tear from under her sunglasses.
French interior minister Bernard Cazeneuve told reporters Saturday that "it seems that he was radicalised very quickly - in any case these are the elements that have come up from the testimony of the people around him", according to Reuters.
Jan Jeffreys and her partner Les Smith, from Shropshire, were enjoying their first evening in Nice when the attack happened.
For several years, extremist groups such as Islamic State and Al-Qaeda have exhorted followers to strike "infidels" - singling out France on several occasions - using whatever means they have at hand.
Speaking outside his home in Msaken, eastern Tunisia, the attacker's father said he had suffered from depression and had no links to religion.