Sharif, in a letter, called on President Hussain to reject appeals of Muhammad Ikramul Haq, Ahmed Ali, Mohammad Tayyab, Ghulam Shabbir and Zulfiqar Ali, officials said.
An official of the President House said the mercy pleas were submitted by the convicts.
The five prisoners, all from Karachi, were convicted of target killings.
Pakistan lifted moratorium on capital punishment a day after Taliban militants killed 150 people, most of them children, in an attack on a military-run school in Peshawar on December 16.
So far, seven convicts have been executed, including those complicit in the attack on former military dictator Pervez Musharraf.
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