Abdullah Abdullah demanded that the preliminary result, due out tomorrow, be delayed for a comprehensive audit of suspicious votes, repeating that he would reject the outcome if his demands were not met.
But Ashraf Ghani, who is reported to be at least one million votes ahead in the count, insisted that the result could not be delayed again and must be revealed by the Independent Election Committee.
The preliminary result will include all votes from the June 14 run-off vote, with the official result scheduled for July 24 after a period for adjudication of complaints.
"What we are asking for is thorough auditing, then preliminary results afterwards," Abdullah told supporters in Kabul.
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"We are honoured to claim that we are the majority of votes in Afghanistan that are clean.
"I will accept the result when the clean votes are separated are from unclean votes."
Abdullah's stance came despite reports that the two candidates were close to a deal agreeing that the audit would be held, but only after the preliminary results.
Influential US Senator Carl Levin, on a visit to Kabul, told reporters that he expected a compromise to be reached.
"Tomorrow there's going to be an announcement of the result," Levin, chairman of the Senate committee on armed services, said.
"Something else is going to be announced... I expect that something else is going to be an agreement on a comprehensive audit.
"We have assurances they will accept the outcome of such an audit."
Ghani's team earlier said that no further delays in the result were unacceptable.
"This is a red line for us," Azita Rafhat, a Ghani spokeswoman, told AFP.
The US and Afghanistan's other international backers had pushed hard for a smooth election as NATO troops pull out, but the fraud allegations have provoked a major political crisis in Kabul.