In a statement issued on Friday evening, SNCF said traffic “would improve” on affected lines on Saturday thanks to the work of thousands of rail workers.
The company added that customers whose trains are delayed or cancelled will be contacted by email or text message.
Eurostar said it expected about a fifth of services over the weekend would be cancelled, while all trains would face delays of around 1.5 hours. Eurostar services use the northern high-speed line.
SNCF said surveillance of the rail network had been strengthened “on land and in the air,” using 1,000 workers and 50 drones.
Junior Transport Minister Patrice Vergriete said around 250,000 people had been affected on Friday, while up to 800,000 could face delays and cancellations by Monday.
He added that disrupting holiday travel, rather than Friday’s Olympic opening ceremony, was the most likely aim of the saboteurs.
“There is not necessarily a link” with the Olympics, he said in an interview.
The last weekend of July is traditionally a busy period for holiday travel.
No group has yet claimed that it was behind the attacks. A source linked to the investigation told the AFP news agency that the operation was “well-prepared” and organised by “a single structure”.
Mr Attal said security forces were searching for those responsible.
At around 04:00 on Friday, saboteurs cut and set on fire specialised fibre optic cables essential for the safe functioning of the rail network, government officials said.
One site was at Courtalain, 150km (93 miles) south-west of Paris. A picture posted online purportedly showed burnt-out cables in a shallow gulley with its protective SNCF paving stones discarded.
The SNCF spoke of a “massive, large-scale attack aimed at paralysing” its services, not just at Courtalain but at Pagny-sur-Moselle, a village outside the eastern city of Metz and Croisilles, not far from the northern city of Arras.
Another attempted attack in Vergigny, south-east of Paris, was foiled by SNCF workers who were carrying out maintenance on site in the early hours of Friday.
Prosecutors have opened an investigation into attacks on “the fundamental interests of the nation”.
