Nothing concrete has been heard about his whereabouts since March 12th, when his car was found abandoned on a Gaza street.
As part of a day of action which includes the reading of the letter at a London press conference and demonstrations by Palestinian journalists, BBC director general Mark Thompson made an appeal for a "safe and speedy resolution to this case".
Speaking at a news conference in Ramallah, West Bank, that was also translated into Arabic, Mr Thompson confirmed that he had been in touch with Palestinian president Mahmoud Abbas.
According to the BBC chief, Mr Abbas said he had "credible evidence that Alan was safe and well" and reassured him that everything that could be done to secure his release was being done by the Palestinian authorities.
"The people of Gaza are ill-served by kidnappings of this nature," Mr Thompson went on, calling on those who may have influence on the kidnappers to help secure Mr Johnston's release.
In the letter read out by Graham Johnston, the family spoke of how they were "hanging in there and trying to keep a stiff upper lip".
"Hello Old Son," the letter began.
"I don't know if you will see this but I just felt that I wanted you to know how distressed and sorry we all are that you were taken.
"You had warned us frequently that the chances were always there that you would be kidnapped and we were prepared in a way for this to happen. Nevertheless when it came it, was still a considerable shock."
The letter thanked the BBC for the support it had provided and also praised the "magnificent job" Palestinian journalists were doing in keeping Mr Johnston's plight in the news with protests and strikes in support of his case.
"I would like to say something to those who are holding you," the reporter's father added.
"You have families. Please think about what this is doing to my family, including in particular the distress and deep, deep concern Alan's mother and sister have had to endure for all these long weeks. As I have said before - please - let my son go, now, today."
Earlier, Mr Thompson confirmed that the BBC had not received any ransom demands from kidnappers and admitted he was "increasingly concerned" about the mental toll Mr Johnston's incarceration is having on him.
When Mr Johnston was abducted he was in the last weeks of his posting and had been "looking forward to returning to his staff post in London" where he was to work at the BBC World Service, Mr Thompson said.
The abduction of Mr Johnston, one of the last remaining western journalists following daily events in Gaza, has been condemned as a "criminal act" by the Palestinian government.
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