Overnight, the President summoned the two candidates Joko Widodo and Prabowo Subianto for talks on how the process should unfold.
Major polling companies which sampled millions of votes have Jokowi holding a comfortable lead.
But his rival Prabowo Subianto claims he has data showing he's won.
The standoff is unlikely to be settled until the electoral commission gives the full official declaration of the poll in two weeks.
Correspondent: Greg Jennett
Speakers: Phillips Vermente, Centre for Strategic and International Studies, Joko Widodo, presidential candidate; Prabowo Subianto, presidential candidate
JENNETT: Jokowi supporters don't need an official declaration to feel or perhaps to know that their man has won.
They gathered in the central Jakarta Park, known as Proklamasi, to celebrate. Crushed and congratulated, Joko Widodo took to the stage.
(Sound of Joko Widodo speaking)
He was softly spoken, as ever but now with a serious political point to make. One by one he read out the conclusions of six polling firms. One was 52-48 in his favour, another 54-46. Jokowi called them "credible", "accurate" and "accepted by all parties."
He was authorising figures to a nation in large part perplexed as to why rival candidate Prabowo Subianto sounded so defiant when the numbers seemed so stacked against him.
SUBIANTO (translated): We are asking all members, supporters of the red and white coalition and all the people of Indonesia to monitor and guard this victory all the way until the official counting is done.
JENNETT: Prabowo relies on the figures of as few as four polling agencies, none of them "big name companies" to support his belief that he's still in this race.
But the big-eight polling companies which called the Jokowi result are adamant there is no doubt about their findings.
Philips Vermonte is from the Centre of Strategic and International Studies which commissioned one of the polls.
VERMONTE: I think it is a very slim possibility for the quick count to get it wrong. And another way of checking it is that there are a number of survey agencies doing the same thing today in Indonesia, I think about 9 or 10, and I think most of the survey agencies got the same numbers.
JENNETT: Joko Widodo isn't waiting for a Prabowo concession. He's basically claiming victory and sounding like a president-elect.
WIDODO (translated): Today a new history was created. This is a new chapter for Indonesia. This victory, based on quick-count results, is not the victory of Jokowi and Jusuf Kalla. It is the victory of all Indonesians.
JENNETT: Indonesia's electoral commission is hoping to deliver an official result on the 22nd of July, two weeks ahead of its month-long constitutional deadline.
Until a result is settled, the coordinating minister for politics, law and security, Djoko Suyanto is appealing to parties and the public to remain calm and not break the law.