The spokesman said its main demand was the release of Abu Qatada, a London-based Palestinian-Jordanian, who faces deportation from the UK.
The British government says Abu Qatada is suspected of links with al-Qaeda and is a "significant international terrorist".
The call for Abu Qatada's freedom was originally made in a video of Alan Johnston which was posted on the internet on June 1.
Negotiations
Abu Osameh al-Mo'ti, a Hamas representative in Iran,had said that the group was in negotiations with those holding Johnston but did not specify how he knew the British journalist would be freed.
"The BBC journalist will be released within the next hours, today," he said.
He also suggested that Hamas knew where the journalist was being held and that he was in good health.
"We are aware of the reports, but have not received any firm confirmation of Alan's situation. We continue to work with everyone involved to try to effect Alan's safe release," a BBC spokeswoman in London said ahead of the Army of Islam statement.
The video was the first sign that Johnston was still alive since he was seized at gunpoint as he drove home from work in Gaza City on March 12.
He said in the video that his captors were treating him well and spoke of the suffering that Gazans had endured because of sanctions imposed on the Hamas-led Palestinian government.
He also criticised the British military presence in Afghanistan and Iraq, saying: "In all this, you can see the British government is endlessly working to occupy Muslim lands against the will of the people in those places."
An Islamic group called the Kataeb al-Jihad al-Tawheed (The Brigades of Holy War and Unity) had claimed in April to have killed Johnston.
However, the Palestinian authorities said at the time that there was no proof that the claim by the little-known group was true.
Bush House, home of the BBC World Service
A spokesman for the Hamas organisation has indicated that the BBC reporter Alan Johnston kidnapped in Gaza on 12 March would be released within hours. 

"The BBC journalist will be released within the next hours, today", said Abu Osameh al-Mo'ti, Hamas spokesman from Iran.

Sami Abu Zuhri, a Hamas official in Gaza was however more cautious about the situation, saying that "There are encouraging indicators that he will be released in the near future. But we cannot determine this in terms of hours." 

Later, a spokesman for the group said to be holding Alan, the Army of Islam denied the release was imminent. The group is demanding that Abu Qatada, described by the United Kingdom government as "significant international terrorist", be released and said that "If they don't meet these demands there will be no release of this prisoner."  

Earlier this month Johnston appeared in a video in which the group holding him demanded that Britain released Muslim prisoners.

The UK Foreign Office described the ongoing confusion as distressful to Alan Johnston's family and friends. The BBC has said that "We are watching developments very closely."