Jamaat-e-Islami party leader Motiur Rahman Nizami, 73, will likely face the gallows in less than a week if the court dismisses his final appeal tomorrow over crimes committed during the country's independence war.
Police said they had stepped up security in Dhaka ahead of the verdict because of fears of Islamist violence, although they added there were no specific threats from any group.
"We've enhanced security in the capital. Officers have been kept on alert," Dhaka Metropolitan Police spokesman Maruf Hossain Sorder told AFP.
The Supreme Court upheld the verdict in January this year, but Nizami's lawyers filed a final appeal.
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Previous convictions of Jamaat officials triggered the country's deadliest violence in decades, with around 500 people killed, mainly in clashes between Islamists and police.
"There will be extra security at the Supreme Court and at important sites in the city," assistant commissioner of police Shiblee Noman told AFP.
The judgement comes as the Muslim-majority nation reels from a string of killings of secular and liberal activists and religious minorities by suspected Islamist militants.
Nizami, Jamaat's leader since 2000 and a former government minister, could still avoid the gallows if he is granted clemency by the president.
Three senior Jamaat officials and a key leader of the main opposition party have been executed for war crimes since December 2013.
Since it was established by Prime Minister Sheikh Hasina's government in 2010, the International Crimes Tribunal has sentenced more than a dozen opposition leaders for war crimes.
Rights groups say the trials fall short of global standards and lack international oversight.