ISLAMABAD, Pakistan — The fate of a Pakistani Christian girl at the center of a contentious blasphemy case hung in the balance on Tuesday, as lawyers applied for her release from jail and an influential Muslim cleric offered his support.

The Pakistani authorities have held the girl, Rimsha Masih, in a high-security jail since Aug. 16, when hundreds of Muslim protesters, angered over claims that she had burned pages from an Islamic holy book, surrounded a police station here in Islamabad to demand that she face prosecution.

Fearing violence, the police filed blasphemy charges against Ms. Masih. Relatives and human rights workers said she was 11 years old and had Down syndrome, and should therefore be exempt from the blasphemy laws. The girl, who comes from an impoverished family of Christian sweepers, was said to have been seen holding a burned copy of the Noorani Qaida, a religious textbook used to teach the Koran to children.

After a brief court hearing on Tuesday morning, Ms. Masih’s lawyer, Tahir Naveed Chaudhry, said a medical board had established that she was 14 years old and had a degree of mental disability. “The report establishes that her mental condition does not match her age and physical condition,” he said.