Abdullah has secured 43.8 per cent of the vote, with his main rival Ashraf Ghani on 32.9 per cent, election officials said, after about 80 per cent of ballots were counted.
If no candidate gains more than 50 per cent, a second- round run-off between the two leading names is scheduled for May 28.
"With the percentage of votes (still to be counted), I suspect there will not be a winner in the first round, but I cannot say decisively," Ahmad Yusuf Nuristani, head of the Independent Election Commission, told reporters.
A second-round vote could be avoided by negotiations between the candidates in the coming weeks, but Abdullah dismissed talks of a possible power-sharing deal.
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"We have not talked or negotiated with anyone about forming of a coalition government," he told reporters after the latest results.
Eight men ran in the April 5 election, with polling day hailed a success by Afghan officials and foreign allies as the Taliban failed to launch a major attack despite threats to disrupt the vote.
Full preliminary results are due on Saturday before the final official result is announced on May 14 after a period for adjudication of complaints.
The 2009 election, when Karzai retained power, was marred by fraud in a chaotic process that shook confidence in the multinational effort to develop Afghanistan and also started a sharp decline in relations with the US.
Turnout from this month's poll is set to be nearly seven million voters from an estimated electorate of 13.5 million -- well above the 2009 figure.
The eventual winner will have to oversee the fight against a resilient Taliban insurgency as 51,000 US-led combat troops leave Afghanistan this year.