Eurostar has told passengers only to travel from Paris to London “if absolutely necessary”.
Its services have been hit by delays with long queues due to industrial action by French customs officers.
Four trains were cancelled on Sunday. The firm has also cancelled three trains on Monday, two on Tuesday and three on Wednesday.
The company says tickets can be changed free of charge, or affected passengers can claim refunds.
“We recommend not to travel unless absolutely necessary, ” Eurostar advised passengers on its website.
“All Eurostar trains are experiencing delays and long queues for journeys from Paris Nord due to industrial action by French customs until March 19th.
“These delays impact our planned timetables and cause subsequent cancellations,” the firm said.
Customs officers are demanding higher pay and better working conditions.
They also want more staff which they say will be needed after Brexit, to help process British citizens who will no longer have European Union passports.
The industrial action is due to last until 19 March.
Passengers have been complaining on social media of long queues in Paris.
Catherine Hope tweeted that it had taken her four-and-a-half hours to clear all the queues.
Another passenger said they had waited for four hours.
Eurostar says the delays are averaging at two hours and they expect similar delays on Monday.
Last week, French unions representing around 17,000 customs workers rejected a government offer of a €14m pay boost, saying it was not enough.