The convention has exposed deep ethnic divisions and challenged the US vision of a strong presidency.
Officials have warned that the whole process would collapse if delegates failed to reach a deal on the country's post-Taliban charter by the end of Sunday.
The Loya Jirga, or Grand Assembly, is entering its fourth week, more than double the time allotted, and has been marred by a boycott from voting on Thursday by nearly half the 502 delegates at the giant white tent on a Kabul college campus.
"If tomorrow we cannot finalise it (the constitution) we will announce to the world that we have failed," said Sibghat Allah Mujaddidi, chairman of the assembly. "We are stuck on one word."
Delegates said that word was "Uzbek", and whether the Uzbek language should be recognised as official alongside Pashto, spoken by majority Pashtuns, and Dari, spoken by Tajiks.
Voting began on Thursday but a
vast number boycotted the polls
The debate has effectively been suspended since Thursday, with behind-the-scenes deals taking up Friday and Saturday.
The scale of the protest has already forced some concessions from supporters of interim leader Hamid Karzai, who insists on a strong mandate to hold his war-weary country together as he heads into elections scheduled to take place in June.
And it has underlined deep seated suspicions between Karzai's Pashtun clan, traditionally at the centre of Afghan power, and smaller groups like the Tajiks, Uzbeks and Hazaras.
Private talks were expected to continue until late into the evening as they had done the night before, with the United Nations and United States acting as mediators.
President Karzai ruled out any
changes to the presidental system
Karzai's opponents want to dilute his powers and bolster those of parliament and the provinces, and some members of the alliance favour a stricter interpretation of Islamic laws than does the more religiously moderate president.
A Western diplomat said Karzai had been forced to give some ground, including making the president "responsible" before parliament and ministers open to censure by lawmakers.