Lawyers for Mohammed Asghar, 71, from Edinburgh, told campaigners at Reprieve they had serious concerns about his mental wellbeing.
He appeared "pale, dehydrated, shaking and barely lucid", and suffering from severe delusions during a recent visit by lawyers, it is claimed.
Mr Asghar was arrested in 2010 in Rawalpindi, near Islamabad, for claiming to be the Prophet Mohammed. He was convicted last week.
His family says he suffers from mental illness and was treated for paranoid schizophrenia in Edinburgh before returning to Pakistan in 2010. They are appealing for him to be released to receive medical help.
Reprieve said lawyers for Mr Asghar finally gained access to their client yesterday and were able to get the necessary documents signed so that he could file his appeal against the sentence.
But they said they had concerns for his mental health and said prison medical staff unaware of his psychiatric illness, were not giving him the appropriate medication.
Maya Foa, director of Reprieve's death penalty team, said: "We are extremely worried about Mr Asghar's mental health, which appears to have seriously deteriorated."
A petition on change.org, addressed to David Cameron and Alex Salmond, is calling for Mr Asghar's release.
The Foreign and Commonwealth Office said it had continuously made representations to the Pakistan government on behalf of Mr Asghar and would continue to do so. The Scottish Government said it was in touch with the Foreign Office about the case.