"Our intelligence services have confirmed to me that he's alive," he said, adding that he knew which group was holding him but would not give details.
The news will come as a relief to Mr Johnston's family following a claim last week that he had been killed by a previously-unheard of group called al-Tawhid al-Jihad Brigade.
In a statement the BBC welcomed Mr Abbas' report, but said that it wants "firm evidence of Alan's well-being and his immediate release".
The 44-year-old was in the last weeks of his posting when his abandoned car was found on a Gaza street on March 12th.
BBC director general Mark Thompson has described Mr Johnston as "our boy" and said that "we all want him home".
Britain's Foreign Office is investigating his disappearance. Since the abduction, protests have been held in across the world, including in the Palestinian Authority and Lebanon. I
In Britain this week more than 1,000 people across the BBC held rallies to highlight his plight.
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