"In connection with a criminal investigation concerning terrorism... 40 persons were taken for questioning. Twelve among them were arrested. The investigating judge will decide on their possible detention later today," the office said in a statement.
"The investigatory results necessitated an immediate intervention. The investigation continues."
Investigators have found links between the Brussels and Paris attackers, some of whom were based in Belgium.
Belgium's federal prosecutor's office said no weapons or explosives were found during the searches, which also involved 152 garage lockups.
A French police couple were stabbed to death outside their home on Monday in an attack claimed by Islamic State. In a video posted on social networks, the Larossi Abballa, the attacker, linked it to the soccer tournament, saying: "The Euros will be a graveyard."
A spokeswoman for Paris public prosecutor Francois Molins said two people close to Abballa would meet a judge on Saturday.
Flemish public broadcaster VTM said the people arrested overnight were suspected of planning an attack in Brussels this weekend during one of Belgium's soccer matches.
Belgian newspaper Le Soir said areas where fans watch matches were potential targets, as well as other crowded areas like shopping centers and stations, citing a security source.
Meanwhile, a Belgian man has been arrested and charged in connection with the Islamic State suicide bombings that killed 32 people in Brussels in March, prosecutors said late Friday.
The 30-year-old, named as Youssef E.A, is one of several charged over the March 22 attacks that struck Brussels airport and a city metro station.
The man has been charged with "participation in the activities of a terrorist group, terrorist murders and attempts to terrorist murders, as a perpetrator, co-perpetrator of accomplice", Belgium's federal prosecutor said.
It added in a statement that several suspects linked to the bombings took part in a reconstruction Friday at an apartment in the Brussels suburb of Etterbeek.
The flat is believed to have been "used as a safehouse" by the attackers and as the point of departure for the attack at Maalbeek metro station, the statement said.
Osama Krayem, a Swede of Syrian origin who has been charged over both the Brussels bombings and November's jihadist attacks in Paris, took part in the reconstruction.
He is suspected of buying the bags used for the Brussels suicide bombings.
Caught on CCTV minutes before the metro bombing alongside suicide attacker Khalid El-Bakraoui, he decided not to go ahead with detonating his own device, according to his lawyer Vincent Lurquin.
Brothers Smail and Ibrahim Farisi, 31 and 27 years old respectively, also assisted with the reconstruction.
They are suspected of renting the flat to the attackers and cleaning it a day after the bombings.
A 31-year-old man arrested in June, identified by prosecutors as "Ali E.H.A" and named by media as Ali El Haddad Asufi, has also been linked to the apartment and took part in the simulation.
Bilal El Makhoukhi, a 27-year-old Brussels resident arrested in April, also took part, although his suspected role in the Belgium attacks is not clear.
He was sentenced to five years in jail in 2014 after going on trial alongside several members of Sharia4Belgium, a group that sent jihadists to Syria.
He is reported to have been injured fighting in the war-torn country.