French government spokesperson Stéphane Le Foll said that France wants the UK-EU exit talks "to begin between now and the end of the year, or early next year."
But Van Rompuy said negotiations were unlikely until a new German government was formed after next September's election.
The talks will be tough but hopefully of mutual benefit, said the former Belgian Prime Minister, adding the UK had to make the "first move".
He said, "Before the German elections and before there is a new German government, I think no serious negotiations will take place. You can always start with more technical matters, but the hard core, the difficult topics, will be tackled after the constitution of a new German government and that will be October/November."
Van Rompuy described the UK's decision to leave the EU as a "political amputation."
Meanwhile, in a European Parliament debate on the future of the EU, dominated by the response to Brexit, Scottish Nationalist Party MEP Ian Hudghton called for the "democratic will" of the people of Scotland to be recognised.
The debate followed a major speech on the state of the union on Wednesday by Commission President Jean Claude Juncker, who made clear that the UK could not expect 'à la carte' access to the single market.
Scotland voted overwhelmingly to remain in the EU in the 23 June referendum on the UK's membership.
Speaking in the debate, Hudghton said, "The EU is not perfect. Juncker accepts that, and I welcome some of the initiatives that he mentioned to tackle challenges such as in the area of job creation and connectivity, and social justice.
"Juncker also said that the EU should empower citizens and embrace solidarity. Well the people of Scotland voted in June to remain in the EU, to constructively continue to work on collective challenges.
"Cross-party efforts are being made right now in Scotland to identify all possible means of protecting Scotland's position, particularly within the single market. The Scottish government is working hard to persuade the UK government to recognise the democratic will of the people of Scotland in formulating its negotiating position, if it ever gets round to that.
"I hope that the Commission, the Council and this Parliament will be open to imaginatively finding a way to reflect the democratic will of people in Scotland. That would be solidarity."
Elsewhere, in an interview with Ouest-France and asked whether he expects the upcoming Brexit talks to be tough, former European Commissioner Michel Barnier, the EU's chief negotiator in the talks, said, "They will be. We're in a different state of mind. If the British have the reputation of having good negotiators, it is possible that Europe has some too."
He added that the UK's vote to leave the EU "reflects the ultra-liberal drift of the EU."