The incidents paralysed several high-speed lines linking Paris to the rest of France and to neighbouring countries, according to Transport Minister Patrice Vergriete.
Vergriete described people fleeing from the scene of fires and the discovery of incendiary devices at the site.
"Everything indicates that these are criminal fires," he said on BFM television on Friday.
As Paris authorities geared up for a parade along the Seine River amid tightened security, three fires were reported near the tracks on the high-speed lines of Atlantique, Nord and Est.
The disruptions particularly affected Paris's major Montparnasse station.
Videos posted on social networks showed the hall of the station saturated with travellers.
French officials denounced the vandalism that stranded thousands of rail passengers. (AP PHOTO)
The Paris police prefecture "concentrated its personnel in Parisian train stations" after the "massive attack" that paralysed the TGV high-speed network, Laurent Nunez, the Paris police chief, told France Info television.
Travel to and from London beneath the English Channel, to neighbouring Belgium, and across the west, north, and east of France was affected by what the French national rail company SNCF called a series of co-ordinated overnight incidents.
Government officials denounced the acts, though they said there was no immediate sign of a direct link to the Olympics.
National police said authorities were investigating the incidents.
French media reported a major fire on a busy western route.
Sports Minister Amelie Oudea-Castera said authorities were working to "evaluate the impact on travellers, athletes, and ensure the transport of all delegations to the competition sites" for the Olympics.
Speaking on BFM television, she said: "Playing against the Games is playing against France, against your own camp, against your country."
She did not identify who was behind the vandalism.
Passengers at St Pancras station in London were warned to expect delays of around an hour to their Eurostar journeys.
Announcements in the departure hall at the international terminus informed travellers heading to Paris that there was a problem with overhead power supplies.
The attack occurred against a backdrop of global tensions and heightened security measures as the city prepared for the 2024 Olympic Games.
Many travellers were planning to converge on the capital for the opening ceremony, and many vacationers were also in transit.
SNCF said it did not know when traffic would resume and feared that disruptions would continue "at least all weekend".
SNCF teams "were already on site to carry out diagnostics and begin repairs", but the "situation should last at least all weekend while the repairs are carried out", the operator said.
SNCF advised "all passengers to postpone their journey and not to go to the station", specifying in its news release that all tickets were exchangeable and refundable.
Valerie Pecresse, president of the regional council of the greater Paris region said "250,000 travellers will be affected today on all these lines".
Substitution plans were under way, but Pecresse advised travellers "not to go to stations".
