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.
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.