The request was filed in Lille and a ruling is expected in 48 hours.
French President Francois Hollande wants to close the Jungle and move the estimated 9,000 migrants it holds to reception centres across France.
The Jungle has become a powerful symbol of Europe's failure to cope with African and Middle Eastern refugees.
The migrants there live amid squalid conditions, many of them hoping to enter the UK illegally by hiding on lorries crossing the English Channel.
President Hollande had said the camp would be dismantled by the end of the year but some aid workers said this week that the demolition could start as early as Monday.
The charities said they were not against the demolition but were concerned that authorities were rushing the process without adequate preparation.
"Contrary to what we were told, the plan is to dismantle the Jungle very quickly and do it in a matter of days," one of the groups applying for the injunction, Secours Catholique, said in a statement.
"It will be impossible to ensure serious preparations that cater properly to people's interests."
But Pascal Brice, the director of Ofpra, the agency in charge of handling asylum requests in France, insisted the closure would go ahead.
"Calais is over, it's finished," he said. "What is at stake is for this message to get across and for people to realise there's no point in going there, that it's a dead-end."
The government has already started moving refugees to new locations in other parts of the country ahead of the demolition.
Figures for the total number of migrants in the camp are disputed. Recent figures released by the French prefecture state that there are between 5,684 and 6,486 people. Aid agencies however, estimate that there are about 9,000 people.
Fresh concerns have also been raised about the hundreds of unaccompanied children in the camp.
French charity, Terre d'Asile, says there are now 1,290 unaccompanied minors at the camp, an increase of more than 400 in two months.
The unaccompanied minors are aged between eight and 17.
However, other aid groups, including Medecins Sans Frontieres, have questioned the accuracy of the survey, stating it was unscientific.

The resulting footage is part of a project using hi-tech scanning methods to explore animals' anatomy.
It took 132 hours for the snake to fully digest the rat, the scientists said. Their work has revealed other strange insights into python digestion.
They presented the study at the Society for Experimental Biology's annual meeting in Prague, Czech Republic.
The researchers carried out a computer tomography or CT scan of an anaesthetised 5kg Burmese python one hour after it had devoured the rat whole.
They also used a technique called magnetic resonance imaging (MRI) to study the creature's internal organs.
By using contrast agents, the scientists were able to highlight specific organs and make them appear in different colours.
A series of MRI images revealed the gradual disappearance of the rat's body. At the same time, the snake's intestine expanded, its gall bladder shrank and its heart increased in volume by 25%.
The researchers, Henrik Lauridsen and Kasper Hansen, both from Aarhus University in Denmark, explained that the increase in the size of the snake's heart was probably associated with the energy it needed to digest its meal.
"It's a sit and wait predator," explained Mr Lauridsen. "It fasts for months and then eats a really large meal.
"It can eat the equivalent of up to 50% of its own bodyweight, and in order to get the energy out of the meal, it has to restart the intestinal system very fast."
The researchers, who are both based at the university's Department of Zoophysiology and the MR Research Centre at Aarhus, say that their approach has several advantages over the "subjective and sometimes misleading" interpretations of dissections.
Dissection induces changes, explained Dr Hansen. "For example, after opening the dense bone of a turtle shell, the lungs will collapse due to the change in pressure.
"And to use these techniques you don't have to kill the animal," he added. "We can do this using live animals and revisit the results over and over again."
The images, they say, will be valuable tools in future studies of animal anatomy for both research and education.
As part of the project, they have produced similarly spectacular images of several other species, including frogs, alligators, turtles, swamp eels and bearded dragons.
Aid groups have asked a French court to delay the closure of the Calais migrant camp, the "Jungle", arguing authorities are not ready to relocate refugees.
[NEXT_CONCEPT]
Scientists have used the latest imaging techniques to look inside a python that had just swallowed a rat whole.