Mohammad Kamaruzzaman of the Jamaat-e-Islami party would be the second senior Islamist to hang for crimes committed during the 1971 war of independence that resulted in secession of the former East Pakistan.
Law minister Anisul Huq told AFP he could be hanged as early as next week unless he sought a presidential pardon.
The 62-year-old assistant secretary general of Jamaat -- Bangladesh's largest Islamist party -- has seven days to petition the president for clemency, Huq said.
Kamaruzzaman was found guilty in May 2013 of mass murder, torture and abductions.
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The Supreme Court on Monday rejected his appeal, triggering violent clashes between police and Islamist protesters, who set off improvised bombs, torched cars and pelted police with rocks.
In the last week Bangladesh's International Crimes Tribunal, a domestic court, has sentenced Jamaat's supreme leader and key financier to death.
Rights groups say the trials have fallen short of international standards.
Independent researchers estimate that between 300,000 and 500,000 people died in the 1971 war.