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The call for Abu Qatada's freedom was originally made in a video of Alan Johnston which was posted on the internet on June 1.
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.