The eight were beheaded in public on Friday in Riyadh for robbery and murder of an Egyptian security guard in 2007, the newspaper reported.
A similar fate could befall the other five Bangladeshis, the Star reported.
"A [Shariah] court has sentenced five Bangladeshis in Riyadh to death in three cases for murdering their three compatriots," the newspaper reported, quoting Haroon-or-Rashid, a Bangladesh diplomat in Riyadh.
The identities of the five or their victims were not available, the report said. Four of the five are in prison and the fifth remained at large, the report said.
Rashid told the newspaper the accused could either file an appeal or reach a settlement with the complainants.
"The four convicts, who have been in jail for four years, could be executed unless the cases are settled amicably," Rashid said. "We have been trying to reach a settlement but have failed thus far."
Amnesty International, commenting on the Friday beheadings, said similar executions have resumed in Saudi Arabia at an alarming rate since the end of holy month of Ramadan.
Hassiba Hadj Sahraoui, the London-based rights group's deputy director for Middle East and North Africa, urged the Saudi government to immediately commute all death sentences "with a view to abolishing the death penalty completely."
AI said the latest beheadings brought the number of executions in Saudi Arabia this year to at least 58, more than double the 2010 figure. Twenty of those executed this year were foreign nationals.
The beheadings were condemned in Bangladesh by human rights activists, the Star reported.
National Human Rights Commission Chairman Mizanur Rahman said the beheadings went against the spirit of international human rights laws.
Sultana Kamal of the rights group Ain O Salish Kendra, said: "There are no words to condemn the execution of the eight Bangladeshis. This is barbaric, appalling and a crime against humanity."