CSAN and Secours Catholique want the French and UK government to urgently look at the fate of many unaccompanied children who are living in the camp.
The first stage of the eviction process got underway on Friday after a judge approved mass evictions.
Volunteers said French officials toured the tents and shacks in the southern area of the site telling people it was time to leave and the Help Refugees charity said around ten adults took up an option to leave on two buses brought to the camp, which is home to around 4,000 migrants and refugees.
Dr Philip McCarthy, CEO of Catholic charity CSAN said, "no one wants to see the camp continue, however we need more time to assess the welfare of vulnerable unaccompanied minors and put in place proper arrangements to prioritise their health and social needs. It is also vital that unaccompanied minors are afforded the opportunity to be reunited with family members who have found sanctuary in the UK or elsewhere in Europe."
Vincent De Coninck, Project Manager in Calais for Secours Catholique, CSAN's partner in France, echoes Dr McCarthy's statement, "the solution to eradicate the camp is not a solution as long as it does not provide proper solutions for everybody."
"We would like the camp to disappear in the future and we're ready to collaborate, but it must take time - we have to discuss it with the refugees, experiment and try different solutions. It's as if the government would like to make people disappear before the spring because they are afraid of new arrivals."
Bernard Thibaud, General Secretary of Secours Catholique adds there is, "a lack of trust between migrants and the government."
Commenting on an appeal before the Calais state council, he warns against the use of any force against, "people who have been victims of violence throughout their entire journey".
Reports from 'The Jungle' claim there have been "lots of conflicting information" given out, including on what accommodation was being offered and for how long.
A judge in Lille ruled on Thursday that a partial clearance should go ahead at the camp, apart from social spaces, including schools and places of worship.
Campaigners wanted to postpone the relocation of people from the slum to heated containers nearby or to centres around France.
State authorities have estimated up to 1,000 people will be affected, however aid workers say the figure is likely to be much higher.
One charity, Help Refugees said its own analysis revealed there were 3,455 people living in the affected area - and Save the Children said nearly 400 unaccompanied children who have fled war, poverty and persecution live in the cold, squalid, rat-infested site.
Mayor of Calais Natacha Bouchart said last week that the dismantling of the camp would keep migrants and refugees away from activists bent on causing disruption.
She said it was a "sensitive situation" that required "necessary firmness," adding that the conditions endured at the Jungle were "unworthy of human nature".
Aid workers are worried that the mass eviction will not solve the problem, merely result in the problems being shifted elsewhere.
Last month around a dozen Methodist ministers held a peaceful vigil at the French Embassy to protest about the conditions in which refugees including families are existing in camps at Calais and Dunkirk.
Protestors held placards saying 'Je suis Refugee', as a mark of solidarity with the refugees whose voices are not being heard.
Revd David Haslam, speaking at the time, said: "Many Christians, and others, have offered hospitality to refugees should they arrive in the UK," said the Revd Haslam. "Surely an agreement could be reached with the French and British Governments to bring in some of those who are only 20 miles from our shores."