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.
Until yesterday no contact had been received from any groups claiming to have kidnapped the Scottish reporter.
In a statement today his family said: "We make a heartfelt appeal to anybody who may have knowledge of Alan's situation and wellbeing to contact the authorities in Gaza.
"Our son has lived and worked amongst the people of Gaza for the last three years to bring their story to the outside world – and we ask every one of them to help end this ordeal."
The BBC said this morning that it still had no independent verification of the rumours of his death.
"We continue to be highly concerned for his safety and are demanding urgent clarification from the Palestinian and British authorities," a statement said.
BBC director general Mark Thompson added that Mr Johnston is "our boy" and "we all want him home".
Britain's Foreign Office has said it is "urgently" looking into the claims, while Palestinian interior minister Hani al-Qawasmi told reporters in Gaza that he had "no information to confirm the killing" of Mr Johnston.
Sunday's emailed statement containing claims about the journalist's death from al-Tawhid al-Jihad Brigade, which translates from the Arabic as "the brigades of holy war and unity", highlighted the number of Palestinian prisoners being held in Israeli jails.
It follows news late last week that Mr Thompson had been told by Palestinian authority president Mahmoud Abbas that there was "credible evidence that Alan was safe and well".
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