Johnston was taken hostage by masked gunmen as he returned to his apartment in Gaza City on March 12.
Speaking on BBC radio, the corporation's head of newsgathering, Fran Unsworth, voiced fears for Mr Johnston's well-being after four weeks in presumed captivity.
Kidnappers have abducted dozens of foreigners in Gaza, but none have been held so long as Johnston.
He is incarcerated, and what that must be doing to his mental state and his general health, we have no idea Fran UnsworthBBC head of newsgathering class="" href="/1/hi/world/middle_east/6524879.stm">Thousands sign BBC petition She has spent the past two weeks in Gaza liaising with officials, and said the local politicians were doing what they could.
Fran Unsworth, the BBC's head of newsgathering, said: "The longer it goes on, the more concerned that we become. He is incarcerated, and what that must be doing to his mental state and his general health, we have no idea."
"The longer it goes on the more concerned that we become," she said.
Lawless territory
"He is incarcerated, and what that must be doing to his mental state and his general health, we have no idea."
In his Easter Sunday plea to the kidnappers, Dr Sentamu said Alan Johnston was not their enemy.
He said Mr Johnston had to be released as a way of ensuring press freedom in the future.
"He is a symbol of ensuring the freedom of the press is not violated, an issue Palestinian journalists indeed are familiar with after facing routine attacks, harassment and arrests.
"And for Alan Johnston, my prayer is that he will be released. After all, he was trying to get the story out."
Cabinet 'determined'
There have been a number of rallies calling for Mr Johnston's release in towns across the Palestinian territories.
Saturday's rallies in Ramallah, Nablus, Jenin and Gaza City were the latest in almost daily protests calling for Mr Johnston's release and urging the Palestinian authorities to do more to find him.
Senior members of the Palestinian government insist that they are doing all they can to locate and free the correspondent.
Senior members of the Palestinian government have repeated their insistence that they're doing all they can to locate and free the BBC reporter.
They have ordered security services and the interior ministry to take "all necessary measures" to secure his release.
They have ordered security services and the interior ministry to take "all necessary measures" to secure his release.
The BBC News website has received thousands of messages of support from readers demanding the release of Alan Johnston.
Johnston joined the BBC World Service in 1991 and has spent eight of the last 16 years as a correspondent, including periods in Uzbekistan and Afghanistan.
The BBC describes Mr Johnston as a highly experienced and respected reporter.
He has lived and worked in Gaza for three years and was the only Western reporter permanently based in the often violent and lawless territory.
He joined the BBC World Service in 1991 and has spent eight of the last 16 years as a correspondent, including periods in Uzbekistan and Afghanistan.
His posting in Gaza was to end at the end of March.