Interior Minister Bernard Cazeneuve said they had information to suggest one of those arrested "could undertake violent actions in France".
But he played down French media reports that an attack was imminent.
France remains on high alert after the jihadist attacks in Paris in November that killed 130 people.
More than 100 people were wounded in a series of shootings and suicide bombings that targeted a concert hall, major stadium, restaurants and bars over the course of a Friday evening.
The so-called Islamic State (IS) group said it was behind the attacks.
French media earlier reported that three men and a woman were detained at dawn in the 18th arrondissement of Paris and the nearby northern department of Seine-Saint-Denis.
They said at least one of those arrested had a prior conviction and had been under house arrest since last month, under new rules imposed after the November attacks.
Mr Cazeneuve confirmed that the authorities had concerns about one of those arrested, saying the suspect was thought to have ties with IS in Syria.
"This person was arrested this morning along with people linked to him," he said.
But he warned against jumping to the conclusion that an attack was imminent, and said such arrests were commonplace.
The Paris prosecutor's office said computer equipment seized during the dawn arrests would be analysed.
The arrests come a day after French police officers, alongside Belgian officers, were involved in a raid on a house in Brussels, that ended in a shoot-out with at least two suspects.
One of the suspects was killed, and later identified as Algerian national Mohamed Belkaid. His body was found alongside Salafist (ultra-conservative Islamic) literature and Kalashnikov ammunition, prosecutors said.
Four people have been arrested in the Paris area as part of a wider investigation into a possible plot against French targets, officials say.