Abdul Rahman has been charged with rejecting Islam, a crime under Afghanistan's sharia or Islamic law.
His trial started last week and he confessed to becoming a Christian 16 years ago. If convicted, he could be executed.
But Sarinwal Zamari, a prosecutor, said questions have been raised about his mental fitness.
"We think he could be mad. He is not a normal person. He doesn't talk like a normal person," he told The Associated Press.
Moayuddin Baluch, a religious adviser to Hamid Karzai, the Afghan president, said Abdul Rahman would undergo a psychological examination.
"Doctors must examine him," he said. "If he is mentally unfit, definitely Islam has no claim to punish him. He must be forgiven. The case must be dropped."
"We think he could be mad. He is not a normal person. He doesn't talk like a normal person"
It was not immediately clear when he would be examined or when the trial would resume.
Some media reports suggest he may not have a lawyer.
A Western diplomat in Kabul and a human rights advocate - both of whom spoke on condition of anonymity - said the government was desperately searching for a way to drop the case because of the reaction it had caused.
Earlier, Gianfranco Fini, the Italian foreign minister, whose country is one of four Nato members with troops in Afghanistan, told Italian television late on Tuesday he had indications the Islamic punishment for apostasy would not be imposed on Abdul Rahman.
No death sentence
"From what I've been told, and I have no reason to doubt it, the death sentence will not be carried out," said Fini, whose ministry had summoned the Afghan ambassador in Rome on Tuesday to discuss the case. He gave no other details.
Abdullah (C): Seeking satisfactory
result from constitutional process
The ruling caused a series of protests in Western states more sensitive to the role of religion in international affairs after the Prophet Muhammad caricatures in a Danish newspaper triggered violent protests and demands for an apology in the Muslim world.
The US, which counts Karzai as an ally, has raised the issue with Abdullah Abdullah, the Afghan foreign minister, urging Kabul to uphold the constitutional right of Afghan citizens to choose their faith.
Nicholas Burns, a senior US State Department diplomat, said at a news conference with Abdullah: "We hope that the Afghan constitution is going to be upheld and in our view, if it's upheld, then of course he'll be found to be innocent."
Abdullah said the US embassy in Kabul had received "hundreds of messages" from Americans and added: "I know that it is a very sensitive issue. I hope that through our constitutional process, there will be a satisfactory result."
"We hope that the Afghan constitution is going to be upheld and in our view, if it's upheld, then of course he'll be found to be innocent"
US State Department
Italy, Germany and Canada all expressed their concern. Germany's top Catholic prelate, Cardinal Karl Lehmann, said Christians should enjoy the same religious freedom in Islamic countries as Muslims did in the West.
The case is believed to be the first of its kind in Afghanistan and highlights a struggle between religious conservatives and reformists over what role Islam should have there four years after the removal of the Taliban.