"This process should start as soon as possible," said German Foreign Minister Frank Walter Steinmeier after a meeting with his counterparts from France, Italy, the Netherlands, Belgium and Luxembourg in Berlin.
"The aim must be "not to fall into a prolonged stalemate", he added.
"We'll start immediately," said French Foreign Minister Jean-Marc Ayrault, adding "we now expect that the process will be triggered under Article 50."
With his colleagues he would send the signal that "Europe is alive,"Ayrault said.
Cameron had announced on Friday his intention to step down in October, after his country has voted to leave the EU.
"I hope that we are not playing cat and mouse," warned Luxembourg's Foreign Minister Jean Asselborn with a view at London's policy.
"The people have spoken. and we need to implement this decision," he said, adding that Britain must now quickly start up the mechanism for exit which was defined in Article 50 of the Lisbon Treaty.
In a related development, Iran's Foreign Ministry announced that the relations between the Islamic republic and Britain will not change after the latter voted to leave the EU, Mehr news agency reported on Saturday.
Iran fully respects the votes of British people on leaving the EU and deems it consistent with the will of the majority of Britain's people in determining their own foreign relations, said Iran's Foreign Ministry statement.
U.S. President Barack Obama has said that he is confident that Britain is committed to an orderly transition out the European Union.
The British "Leave" camp has won the historic referendum held on Thursday by obtaining nearly 52 per cent of ballots, pulling the country out of the bloc after 43 years of membership.
Gianni Pittella, President of the Progressive Alliance of Socialists and Democrats (S&D) group in the European Parliament said that the Brexit vote was used to reshape the internal political balance in Britain.
"A very serious topic of membership to the EU was used to redesign the internal political balance in UK and this is unacceptable," he said.