"They submitted two petitions to the Ministry of Home Affairs through the jail authority, and the ministry has now forwarded those to the Law, Justice and Parliamentary Ministry," Law Minister Anisul Huq told Reuters news agency.
The two leaders have already exhausted all legal appeals to avoid execution and their fate now rests with Bangladesh President Abdul Hamid, who has the power to pardon or commute death sentences.
Chowdhury, 66, is an ex-lawmaker and a top aide to Khaleda Zia, leader of the main opposition Bangladesh Nationalist Party. Mujahid, 67, is the second most senior member of Bangladesh's largest Islamist party, Jamaat-e-Islami.
Both are among more than a dozen leaders of the opposition alliance convicted by the tribunal, which was set up by the secular government in 2010 and criticized by rights groups and the United States.
Human Rights Watch said the tribunal allowed the prosecution to call 41 witnesses, while Chowdhury's defense was limited to just four witnesses. The New York-based rights group said Mujahid was sentenced to death for instigating his subordinates to commit abuses, although no subordinates testified or were identified.
Meanwhile, US lawmakers overseeing foreign policy described the process as "very flawed" and a means of political retribution. The State Department was less forceful but said Friday that executions should not take place until it is clear the trial process meets international standards.
Both men were convicted for crimes during Bangladesh's bloody war of secession from Pakistan in 1971. Military leaders in the west of Pakistan responded to a crushing election victory by the east-based Awami League party - then led by current Prime Minister Sheikh Hasina's father - with a bloody crackdown.
That culminated in a war with neighboring India that Pakistan lost, leading to the creation of Bangladesh from what once was East Pakistan.