"We won't make an exception for anyone. If you need money to release our prisoners we will give you all you need up to the last dirham we have."
This is the first time demands have been made by Johnston's purported kidnappers since he was believed to have been seized at gunpoint on his way home from work in Gaza on March 12.
Nour Odeh, Al Jazeera's Gaza correspondent, said there was no information on Johnston's whereabouts, his health or the group's intentions, but noted that the absence of any threat could be a good sign.
She said the group's demands were unusual as kidnappings in Gaza commonly had to do with factional fighting or the conflict with Israel.
She added that Jaish al-Islam was not a group seen on Gaza's streets and only came to prominence last year when it participated in the capture of Israeli Corporal Gilad Shalit alongside the more established Hamas and Popular Resistance Committees.
Britain's foreign office said on Wednesday it was urgently investigating the tape.
"We are aware of that a video containing images of Alan Johnston, the British journalist abducted in the Gaza Strip has been released to the press," the foreign office said in a statement.
"A video such as this is very distressing to Alan's family, friends and colleagues. We are of course in constant contact with his family and with the BBC.
"We are looking into this as a matter of urgency. In particular, we are continuing to work extremely closely with the Palestinian Authority in order to establish the facts and to try and secure Alan's safe release."
The BBC told AFP that the recording was not a video tape, but an audio tape accompanied by a still image of Johnston's BBC identity card.
Mustafa Al Barghouti, the Palestinian information minister, told Al Jazeera the unity administration was still working for the reporter's release.
"We are getting close to having him released without financial and political blackmail being conducted.
"What has happened is a criminal act - the people who kidnapped him behaved like a criminal mafia.
"Our aim is not only to release Alan Johnston but make sure that similar kidnappings don't happen in the future.
"Thats why his release is very much related to our ability to implement our security plan."
Ismail Haniya, the Palestinian prime minister, last week said his aides were seeking a possible meeting with people close to those involved in the kidnapping.
An unknown Palestinian group had claimed last month that it had killed Johnston but Mahmoud Abbas, the Palestinian president, said a few days later that he was still alive.