As of Sunday, less than a quarter of the ballots had been counted in the Central African nation.
Unless one of the 33 candidates wins the majority, the two top vote-winners will face each other in a runoff.
With 4.7 million votes counted and certified, Kabila has received 51% of the vote, compared with 19% for Jean-Pierre Bemba, an ex-rebel leader and vice president in Congo's postwar national-unity government.
The election - Congo's first multiparty presidential vote in more than four decades of war and unrest - is seen as a chance to establish democracy and peace.
Kabila was seen as the front-runner because many in Congo credit him with taking the initiative to end Congo's 1996-2002 war by uniting warring rebels to form a transitional government that paved the way for the elections.
But some are also suspicious that he is being forced on them by the international community.
In all, about 80% of Congo's 25 million registered voters cast ballots in the presidential and legislative election. Congo has 58 million people.
A preliminary countrywide tally is expected to be announced on August 20, and a final tally on August 31.
The latest count includes about a million more ballots than were tallied on Saturday, which had put Kabila's share at 53%, helped by a blizzard of ballots from the east where Kabila's support is substantial.
Many of Sunday's additions came from the Equateur region where Zanga Mobutu, son of Congo's last authoritarian ruler, Mobutu Sese Seko, put up a strong showing.
Kabila's support is strong in the
Central African nation's east
Mobutu has received at least 2.7% of the vote, though a more specific figure was not available.
Others who have received at least 2% of the vote include Pierre Pay Pay, a longtime ally of Mobutu Sese Seko, Harvard-educated Dr Oscar Kashala and 80-year-old Antoine Gizenga, most showing strength in their native provinces.
No firm results have yet been posted from the capital, Kinshasa, where Bemba is believed to have strong support among the city's eight million people.
The prospect of Congo's population rejecting the vote troubles foreign diplomats, who say the elections' legitimacy is vital for getting Congolese behind their first democratic leader since 1961.
Many international observers noted irregularities in the voting and protracted and chaotic counting, but none so far considered serious enough to affect the outcome.
Both Bemba and Azarias Ruberwa, another vice president and former rebel leader running for president, have alleged fraud in the vote, though both have said they will accept results if they are deemed fair by international observers.
Bemba has said he is leading Kabila in many parts of Congo.