It comes as ferry firms entered talks with port authorities after travellers faced queues of up to 17-hours.
However, speaking to the BBC, Eurotunnel said it was unlikely to have availability as bookings were made in advance.
Eurotunnel spokesman John Keefe reportedly said the service, which experienced "no delays" last weekend, was already running at maximum capacity.
"We have a limited number of trains and four is the maximum we can operate per hour. Also we have truck shuttles, Eurostar and freight trains. It’s a very intense environment."
Travellers were warned on Wednesday (5 April) of further cross-channel delays over the Easter bank holiday weekend.
