Former police commando Mumtaz Qadri, deputed on the security of Taseer, had killed him in Islamabad in 2011 for allegedly criticising blasphemy laws.
The self-confessed killer was arrested and was handed down death sentence the same year by an anti-terrorism court.
His appeal against the sentence were successively rejected by the Islamabad High Court and the Supreme Court last year.
The final review petition was also thrown out by the Supreme Court in December, 2015, leaving the killer with the last option of filing a mercy appeal with the President who in certain cases can condone the death sentence.
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With the decision on the mercy petition pending with the President, the security of his family has been strengthened fearing a backlash if he rejects the petition.
The President has three sons. Some members of his immediate family live in Karachi, his home city, but they have been moved to the President's house in the capital, officials said.
Hussain's own security has also been heightened, they said.
Only family of the victim can forgive a killer but in crimes against state like the killing of a governor, the head of the state can also intervene.
Radical religious groups have been demanding that Qadri should be forgiven as he killed a "blasphemer".
Taseer, who died aged 66, had termed the blasphemy regulations as "black laws" drawing the ire of extremists.
The laws were introduced by Pakistan's military ruler Zia-ul-Haq in 1980s.