A spokesman for Van Rompuy told the AFP news agency that he had been officially proposed by Sweden, which currently hold the EU presidency, to become president.
Dirk De Backer said: "The Swedish presidency has proposed Mr Van Rompuy to be president of the European Union."
An EU diplomat, speaking on condition of anonymity, said Van Rompuy now "seems certain" to take the role.
Hamish Macdonald, Al Jazeera's correspondent in Brussels, said Van Rompuy is seen "as a low-key, conciliatory option".
Jean-Claude Juncker, Luxembourg's prime minister, is also in the running for president, as is John Bruton, a former Irish premier.
Experts have said the president should be a technocrat who can provide unity and build consensus among the EU's main institutions - the council of nations for the 27 member states, the European Commission and the European parliament.
Glyn Ford, a former Labour MEP, told Al Jazeera that building a political presence in Europe was one of the challenges a new president would face.
"Europe is bigger than the United States, its richer than the United States, it gives more to the developing world than the United States by far. But we don't actually have the political presence," he said.
"Now how do you develop that political presence?"
Earlier, Fredrik Reinfeldt, the Swedish prime minister, appealed to leaders to keep the horse-trading to a minimum.
"I need of course the collaboration of my colleagues to try to get this through," he said.
The summit "might take a few hours, it might take all night", Reinfeldt said.
Angela Merkel, the German chancellor, who has pledged to push for a common candidate along with Nicolas Sarkozy, the French president, said on Wednesday that she was confident a candidate would be agreed on by Thursday night.
"I'm optimistic that we will have an agreement [Thursday] evening," she said.