The Army of Ansar al-Sunna issued a statement on its web site on Sunday denying reports that it had killed Hassoun.
"This statement that claimed to be from us has no basis in truth...and we have an official Web site through which we publish statements," the group said, referring to a statement issued on two Islamist Web sites on Saturday in the name of Ansar which said the armed group had beheaded Hassoun.
However, the group said it believed that killing "such filth brings one closer to God".
On 27 June Aljazeera aired a video showing Wassouf being held by a group calling itself the Islamic Retaliation Movement/Armed Resistance Wing.
The captors had warned they would carry on attacking Iraqi officials and US-led foreign forces, despite Washington's handover to an interim Iraqi government last Monday, until "God's law" prevailed in Iraq.
Earlier, a source at the ministry said on Sunday the Lebanese consul in Baghdad, Hasan Hijazi, told Beirut on Saturday that Hassoun, an American of Lebanese origin, had been beheaded, but gave no further details.
But after his strong condemnation of the alleged killing, Lebanese Foreign Minister Jean Ubaid told Aljazeera that news of Hassoun's death "was not official."
"This statement that claimed to be from us has no basis in truth...and we have an official Web site through which we publish statements"
The Army of Ansar al-Sunna
Ubaid said the Lebanese charge d'affaires in Baghdad was "in contact with some forces that have indirect links to the (captors), and these forces say they lost hope in all attempts and by last night the group was about to behead him or had already beheaded him."
"This act is irreligious, immoral and inhuman," Ubaid said earlier in comments carried by Lebanon's national news agency.
There was no immediate confirmation from the US military in Baghdad.
Family awaiting news
Hassoun has been absent from
his unit since 21 June
Hassoun's family in northern Lebanon told Reuters it had not received confirmation of the killing.
"Until now, there is no confirmation," his brother Sami Ali Hassoun said. "We are trying to find out the truth."
The US military has said Hassoun had been absent from his unit since 21 June. His Lebanese father had urged his son's captors to have mercy on him as a Muslim and an Arab.
The US military says Ansar al-Sunna could be one of a handful of groups carrying out attacks in Iraq.
The group claimed responsibility for human bombings last February in the northern Iraqi city of Irbil that killed more than 100 people.