A group calling itself Jaish-e-al-Islam (The Army of Islam) claims to be behind the tape, which makes a number of demands and includes an apparent image of Mr Johnston's BBC identity card.
Demands listed in the recording feature a call for Britain to release Muslim prisoners, including the Islamic preacher Abu Qatada, who is currently facing deportation to his native Jordan.
"We demand from Britain that it release our prisoners and particularly Sheikh Abu Qatada the Palestinian," the tape says, according to an extract reported by al-Jazeera English.
It also warns that Muslim prisoners held in "infidel" countries elsewhere have not been forgotten and warns other nations that they could face retaliation if they do not release them.
If the authenticity of the tape is confirmed, it will be the first time that public demands have been made by the kidnappers of Mr Johnston, who was seized at gunpoint on March 12th as he made his way home in Gaza City.
Last month a previously unheard of militant group claimed that it had executed the 44-year-old reporter, who was the only western journalist to be permanently based in the troubled region.
However the claim from the Tawhid and Jihad brigades could not be verified.
Yesterday Palestinian prime minister Ismail Haniya met with British consul Richard Makepeace in Gaza to discuss Mr Johnston's kidnapping.
Mr Haniya's deputy Azzam al-Ahmad said following the meeting that the Palestinian government hoped for a "quick resolution" to the case.
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