Emmanuel Macron said the group will be ordered to be dissolved on Wednesday, when a mosque that relayed a denunciation of the high school teacher is also to shut.
Speaking after a meeting with regional officials working to counter radical Islamists, Mr Macron added that other associations and individuals are on the radar to be shut or silenced.
A judicial official said early on Wednesday that seven people detained as part of the investigation into the grisly murder, including two minors, were to go before a magistrate later in the day for preliminary charges.
The seven were among 16 people, including five adolescents, initially detained for questioning. The other nine were being released.
Meanwhile, more than 1,000 people gathered in drizzly rain to honour Samuel Paty where he was beheaded on Friday as he left school in Conflans-Sainte-Honorine, north-west of Paris.
Mounds of bouquets of flowers were piled in front of the school.
A terror investigation is under way into the murder by an 18-year-old Moscow-born Chechen refugee, who was later shot dead by police.
The killer has been identified by authorities as Abdoullakh Anzorov.
Mr Paty had shown caricatures of the prophet of Islam to his class earlier this month.
His civics course led to parental complaints and threats.
Sixteen people were detained, including members of the killer’s family and five young adolescent students at Mr Paty’s school.
Investigators are trying to learn how the killer, who lived in the Normandy town of Evreux, set up his encounter with Mr Paty, whether there was complicity and whether the beheading was premeditated.
Speaking in the Seine-St-Denis region, north-east of Paris, Mr Macron reiterated that he wants “tangible results” to combat “an ideology of destruction of the (French) Republic”.
Mr Macron said a group called the Collective Cheikh Yassine will be ordered to be dissolved at Wednesday’s Cabinet meeting.
Named after a killed leader of the Palestinian organisation Hamas, the group was founded in the early 2000s by a man who is among those detained for questioning.
Mr Macron did not provide details on how the group was “directly implicated” in the attack.
Interior minister Gerald Darmanin said later on BFMTV that the person in question helped disseminate the virulent message of a student’s father against the teacher.
Mr Macron has asked for quick, concrete action in the case.
The French president is waging war on what he calls “separatism”, referring to Islamist extremism that authorities say has created a parallel world in the country which counters French values.
A mosque in the north-east Paris suburb of Pantin is also being closed for six months starting on Wednesday night.
A sign posted by the regional prefecture at the mosque entrance said the house of worship would be closed for six months – with a six-month prison sentence for violators.
The Pantin mosque is being punished for relaying a message on social media from the father of a student with a virulent complaint about Mr Paty.
The father quoted his 13-year-old daughter as saying the teacher had asked Muslims to leave the classroom – a version that was contested by Mr Paty himself, according to press reports.
Authorities say the mosque has long had an imam following the Salafist path, a rigorous interpretation of the Muslim holy book.
Pantin was also the home of an 18-year-old Pakistani refugee who three weeks earlier attacked and injured two people with a meat cleaver.
A national memorial event will be held on Wednesday evening to pay tribute to Mr Paty in the courtyard of Sorbonne university, a centuries-old symbol of the “spirit of Enlightenment” and “a forum to express ideas and freedoms”, the French presidency said.