Bangladesh's opposition party, Jamaat-e-Islami, has called for a nationwidewide shutdown following the arrest of its leader, local media have reported.
AKM Yusuf was arrested on Sunday shortly after an order by a tribunal for suspected war time abuses.
Al Jazeera's special correspondent, reporting from the capital Dhaka, said that there had been some vandalism following the call for Tuesday's protests, but that the streets were relatively quiet.
Meanwhile, Ban Ki-moon, the UN chief, called for rival Bangladeshi political parties to calm tensions ahead of an upcoming election.
In a meeting with Bangladeshi Foreign Minister Dipu Moni, Ban also "reiterated his increasing concern about the recent wave of violence" linked to Islamic fighters in Bangladesh, said UN spokesman Martin Nesirky.
Ban "stressed the critical importance for the political leaders of Bangladesh to engage in constructive dialogue, with a view to defusing tensions and resolving differences, including on governance arrangements during the upcoming election period," said Nesirky.
The ruling Awami League and opposition Bangladesh Nationalist Party (BNP) are on a collision course ahead of a national election to be held in January 2014.
Extensive coverage of war crimes trials and calls for blasphemy laws that have divided the nation.
The BNP and its smaller Islamist allies have threatened to boycott the polls if they are not held under a neutral caretaker government.
The government of Prime Minister Sheikh Hasina has rejected the demand.
The atmosphere has been further poisoned by clashes between police and Islam protesters last week.
Dozens have been killed in those protests which started days after a court death sentence order against another Jamaat leader, Muhammad Kamaruzzaman, who is accused of masterminding the killing of at least 120 farmers during the country's 1971 independence war.
Ban has sent an envoy, assistant secretary general for political affairs Oscar Fernandez-Taranco, to try to bring the rival parties together.
Speaking in Dhaka, Fernandez-Taranco highlighted the "significant increase" in violence in recent weeks, including the clashes between police and Islamist protesters.
He said: "The views that I have heard suggest that there is ground for agreement but bridging the gap will require political will and commitment to resolve the remaining differences."