Haq's brother, Daoud Arsalan, said that they had now accepted the "sad truth" that Haq had been "martyred by the Taleban".
Taleban spokesman Mohammed Tayyab Agha had earlier told the BBC that Haq and two other men had been killed after US attempts to rescue him failed.
He said at least one American had been travelling with Haq and was now on the run.
The Pentagon said it could not confirm Haq's death nor any rescue attempt, but said the death would be a loss for the prospect of a broad-based government in the country.
Haq, 43, one of the best-known guerrilla commanders who fought against the Soviet occupation of Afghanistan in the 1980s, is thought to have slipped back into the country from Pakistan with the aim of raising a rebellion against the Taleban.
Omar Samad, director of the Afghanistan Information Centre and advisor to the US Congress, told the BBC that he thought Haq had been betrayed by someone linked to the Taleban before his departure from Pakistan.
An ethnic Pashtun, like the Taleban, Haq had been critical of the current US bombing campaign against Afghanistan, saying it could damage his attempts to win over moderate elements within the Taleban.
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Taleban forces said they captured Haq after surrounding his hiding place in the province of Logar south of the capital, Kabul.
Haq was executed at 1300 local time (0830 GMT) on a religious edict issued by Mullah Omar and Muslim clerics in Afghanistan, Mr Agha said.
Haq's nephew, Mohammed Yousuf, said that he his uncle and a companion named only as Hamid were hanged following their capture.
In other developments:
Meanwhile American warplanes continued to bomb the Afghan capital, Kabul, on Friday - the Muslim day of prayer.
Reports said at least 10 bombs fell in quick succession during night raids and big fireballs could be seen north of Kabul.
The attacks followed a daylight raid on the city in which storage facilities belonging to Red Cross workers were struck, which the organisation said were clearly marked.
"I don't personally believe that we are being bogged down or are getting bogged down," said spokesman Rear Admiral John Stufflebeem. "This is a very complicated operation. This is not traditional force-on-force warfare."
He said that although there might be differences between the aims of the US and the Northern Alliance, they were offering each other mutual support.
Taleban Supreme Leader Mullah Omar remained defiant, calling on the movement's supporters around the world to demonstrate against what he called
the oppression of Muslims by countries such as the US and
Russia.
In a pre-recorded statement played to the BBC by Taleban
officials, Mullah Omar said it was not terrorism, but the causes of
terrorism which were responsible for an unstable world. He accused
the US of using the media to label the holy war, which he
said was the duty of all Muslims, as terrorism.
The UK Government earlier announced that it was to deploy ground troops to help US-led operations.
A force of 200 commandos will operate from warships off the coast of Pakistan, with another 400 on stand-by in the UK.
Mr Agha said US helicopters had launched a dramatic rescue attempt after Haq called for help by satellite phone, but he was caught as he tried to flee on horseback.
The Pentagon denied that the US military was getting bogged down in Afghanistan with no victory in sight.
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