French Interior Minister Christophe Castaner said there were clear "failings" monitoring the man who killed four of his colleagues last week.
"Obviously there were failings because three men and one woman are dead," Castaner told a French evening news bulletin on TF1 Television on Sunday.
The 45-year-old administrative worker who committed the attack in Paris' police headquarters previously had an argument with a colleague in which he justified the attack on the offices of the Charlie Hebdo newspaper in 2015 which killed 12 people.
Castaner told TF1 on Sunday that the attacker's colleagues had warned police in July 2015 about the altercation, but that they decided not to file a report about the incident.
France's anti-terror prosecutor Jean-François Ricard said on Saturday that the attacker in the Paris police headquarters had followed a "radical vision of Islam".
In an interview with French radio station France Inter on Monday, Castaner said the main flaw of the system is the lack of an automatic reporting system.
He told the radio station that since 2013, 59 terror attacks have been thwarted—three since the beginning of the year.
"Today in France, the risk of terrorism remains very high, and so we need to give ourselves every means to tighten the sieve to the maximum," Castaner said.
In an interview published in the Journal du Dimanche newspaper, French Prime Minister Edouard Philippe announced the launch of two missions to detect signs of radicalisation - including one centred in the intelligence directorate of the police headquarters where the attacker worked.
"It is our responsibility to never accept any potential deficiencies and to always close the cracks," Philippe said.
French counter-terrorism measures have come under scrutiny by human rights groups who have said that emergency powers, that were first declared after the November 2015 attacks and later codified into a 2017 security law, were used to unjustly punish people.
Meanwhile Castaner is under pressure from right-wing politicians to resign after the attack occurred in the heart of the French capital with opposition parties calling it a "state scandal".
He is due to be questioned in private by the parliamentary intelligence delegation on Tuesday.