Mohammad Asghar was arrested in 2010 for alleged blasphemy and sentenced to death in January. He was in Adiyla Jail in Rawalpindi as he has filed an appeal against his conviction in a High Court.
The Briton of Pakistani-origin was shifted to a hospital after the attack.
According to reports he is facing some health complication linked to old age, his mental instability and gunshot wound.
Blasphemy is punishable by death in Pakistan. Blasphemy laws were introduced by former dictator Ziaul Haq in 1980s. They are sometimes misused by the people against opponents including both Muslim and non-Muslims.
So far authorities have failed to stop the misuse due to pressure by extremists.
Those accused of blasphemy are often lynched and lawyers in defending those accused of blasphemy cases have frequently been attacked.