Afghanistan's first democratic transfer of power was thrown into turmoil when Abdullah said the Independent Election Commission (IEC) was guilty of fraud and he considered it illegitimate.
The dispute threatened to tip the country into political instability at a sensitive time, with the withdrawal of US-led combat forces after 13 years of fighting Taliban insurgents.
The UN had been reluctant to interfere in the election, but outgoing President Hamid Karzai said he would welcome its help as the deadlock deepened.
"UNAMA will continue to support the electoral process that remains fully in the hands of Afghan electoral institutions and stakeholders."
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A potential breakthrough emerged yesterday when senior IEC official Zia-ul-Haq Amarkhail, whom Abdullah had accused of fraud, resigned.
But a smooth election process still appears in doubt, with Abdullah and his poll rival Ashraf Ghani both confident of victory as vote counting continues after the run-off election 10 days ago.
According to reports, Ghani has made a surprise comeback and is ahead in the vote count after finishing well behind Abdullah in the first-round election on April 5.
The preliminary result is due on July 2 and the final result, after adjudication of complaints, is scheduled for July 22.