The UK is due to assume the EU Council presidency in mid-2017, but this has been cast into serious doubt by the decision to leave the EU.
At a summit in Brussels, Cameron is reported to have told other leaders it will be up to his successor to decide if the UK would still take up the presidency, which rotates between member states every six months.
If Britain does not assume the presidency, it is possible that Malta, which has the EU Council presidency from January 2017 for six months, could be asked to continue for another six months.
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Another possibility believed to be under discussion is that Estonia's presidency, due to start in January 2018, could be brought forward to fill the void left by the UK.
Meanwhile, Prime Ministers and leaders of the Party of European Socialists (PES) have agreed that there should be no delay before the UK begins the process to leave the EU.
Urging the UK and EU to act "calmly and responsibly", the leaders agreed that following Cameron's resignation a replacement "must be selected as soon as possible" so as to start the negotiation process at the earliest possible date.
Cameron himself said a successor is not likely to be in place until September or October, when the Conservative party conference is held in Birmingham.
Sergei Stanishev, former Bulgarian Prime Minister and PES president said: "23 June marked a political earthquake in the EU's history, but we must now look forward. Our responsibility, as the progressive political family, is to take leadership during this historical moment for Europe and set a new agenda for the EU.
"We must show that our people-focused policies are more relevant than ever and that we are committed to leading, and bringing change, for the future of Europe."
He added, "We must come to a settlement with the UK as quickly as possible so that we can focus on the many challenges the EU faces - fostering sustainable growth, creating jobs and solving the migration challenges. We must address those to give a decent future to young Europeans."
The PES has drafted a proposal for a 'Social-Democratic Roadmap for Europe', that will be presented on its next leaders' meeting in Paris on 7 July.
Elsewhere, Commission President Jean-Claude Juncker expressed his "astonishment" that the Leave camp had "no clear plan" for the way out of the current political turmoil in the UK.
Speaking in Brussels on Wednesday, he said, "I thought that those wanting to leave would have a plan, a project or a global vision. I can't understand this."
However, Juncker said he understood Cameron's request for "some time to think, because he was Remain campaigner."
Also speaking at the Brussels summit, Belgian Prime Minister Charles Michel said, "Some think that Britain needs more time. I hear this, yes, but I find it strange. It's a type of surrealism."
More comment came from Greenpeace EU deputy director Saskia Richartz, who said, "Whatever the political path forward after the UK's referendum, it's time for EU leaders to think beyond the common market and build a better society that meets the needs of people and the environment."
At the end of the summit, EU leaders issued a statement, saying, "We deeply regret the outcome of the referendum in the UK but we respect the will expressed by a majority of the British people. Until the UK leaves the EU, EU law continues to apply to and within the UK, both when it comes to rights and obligations."