The progress during a one-day break at the constitutional grand council, or loya jirga, could avert a complete collapse of the gathering, seen as a historic opportunity to help this war-shattered nation toward peace and stability.
At the 502-member convention, President Hamid Karzai's fellow ethnic Pashtuns have swung behind his call for a strong presidency, which he says is needed to hold the country together as it rebuilds and faces a stubborn Taliban insurgency.
But smaller groups, including Tajiks and Uzbeks from the north, have dug in against a charter they claim would bring back the Pashtun hegemony that was eased with the Taliban's defeat two years ago.
Exasperated council leaders were forced to abandon voting on the charter in a tumultuous session Thursday, after more than one-third of the delegates boycotted the ballot and held a sit-in in the huge tent where the three-week-old meeting is taking place in the capital, Kabul.
But after private talks Friday with U.N. Special Representative Lakhdar Brahimi, U.S. Ambassador Zalmay Khalilzad and council leaders, rebel delegates told The Associated Press there was new hope that the constitution could be ratified soon.
Mohaiuddin Mahdi, a Tajik delegate from Kabul, said the U.S. ambassador had called his group for talks possibly to discuss a deal to let Afghan officials, who've returned from exile to hold on to their U.S. passports, at least temporarily.
Mahdi said most of the holdouts' demands had been met, including recognition of Uzbek as an official language alongside Pashto and Dari -- a request which has been dismissed by many Pashtuns and has poisoned the debate.
"Our proposals are national proposals, not for one tribe, one province or one ethnic group," Mahdi told The Associated Press. "After today, there is no danger of failure."
Khalilzad has been an energetic presence at the convention, cajoling delegates on the margins and meeting privately with key players to press forward a state-building exercise that could provide lessons for Iraq.
But with the strong presidency sought by Karzai, who enjoys Washington's firm support, apparently already settled, aides are keen to dispel any impression that he's steering developments.
A U.S. Embassy spokeswoman confirmed Khalilzad's involvement in Friday's shuttle diplomacy, but declined to give any details.
Still, Kayum Karzai, one of the president's brothers and a delegate at the convention, also said that an agreement was now within reach.
"Yesterday evening and today, there have been some very good developments," he told the AP. "Hopefully, we can go to the jirga tomorrow and present a united front."
He said government supporters were ready to accept a demand for a constitutional commission to oversee the implementation of the new basic law.
Members of the commission would be "selected by the president and endorsed by the parliament" under a redrafted article, he said.
A key article in the draft has already been amended to state that no future legislation can run counter to the "provisions" of Islam seen by some as code for Islamic Shariah law.
That decision could further alarm human rights groups, who worry that religious hard-liners already in control of the Supreme Court could win an even firmer grip on the judiciary in return for supporting the presidential system.
Council leaders and Western diplomats suspect that the leaders of the armed factions, who still control much of the country, may be fanning the debate over emotive ethnic issues in order to secure powerful positions in a constitutional court or as a vice president.
"The groups who are meeting [with U.S. and U.N. officials] are not ethnic groups, they are groups of like-minded people,'' Kayum Karzai said. "It is open to interpretation, and some people are using the wrong word."
Meanwhile, U.S. troops and helicopters killed as many as 14 insurgents in clashes in the mountains of eastern Afghanistan, the military said Thursday.
Three U.S. soldiers were wounded in the battle Wednesday some 20 kilometers (12 miles) northeast of Shkin, a town in Paktika province near the Pakistan border.
The first three militants were killed in a gunbattle after a small group of insurgents fired on a U.S. patrol, spokesman Lt. Col. Bryan Hilferty said.
The same patrol later found the insurgents, and AH-1 Cobra attack helicopters flying in support inflicted several more casualties, Hilferty said.
"We don't know for sure, but we think 11," he said. "It was getting night and we didn't go to see the bodies."
One of the wounded American soldiers was evacuated and is in a stable condition, Hilferty said. The other two immediately returned to duty.
Suspected Taliban and al-Qaeda militants regularly attack U.S. and allied Afghan forces as well as government and aid workers in a broad swath of southern and eastern Afghanistan along the rugged Pakistani frontier.
The U.S. military, which still has more than 11,000 troops here, says it killed 10 militants and detained more than 100 people in a four-week operation in the border regions called Avalanche which ended Monday.
Afghan, U.S. and U.N. officials have urged Pakistan to prevent insurgents from using the country as a base for cross-border attacks.
Pakistan, a key ally in the United States' war on terrorism, insists it is doing all it can.