Imdad Ali, who is aged around 50, was sentenced to death for the murder of a religious cleric in 2002.
He had been scheduled to hang on September 20 in a prison in the city of Vehari despite having been diagnosed with schizophrenia.
Ali received a last-minute stay of execution from the Supreme Court last week. But with that stay now expired, he could receive a new "black warrant" and face execution as early as next Tuesday.
"It is indisputable that Imdad suffers from serious mental illness," said Harriet McCulloch, deputy director of the death penalty team at international charity Reprieve.
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"There is therefore no doubt that, should Pakistan execute him, it will be committing a grave violation of both Pakistani and international law.
"It is shocking that the system has failed Imdad at every turn -- right the way up to the Supreme Court. The Pakistan government must immediately halt Imdad's execution, and undertake a comprehensive review into how someone who is clearly mentally unfit to be executed has been allowed to come so near to the noose."
Hangings were initially reinstated only for those convicted of terrorism, but later extended to all capital offences. The number of those executed since then now stands at 419 from more than 8,000 death row prisoners.
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