The group, which calls itself the Youth Shadow Government, said Mubarak should be given the death penalty as punishment for those who lost their lives during the 30 years he ruled Egypt, al-Masry al-Youm reported.
The ailing former president is currently on trial in Cairo Criminal Court on charges of killing protesters, squandering public money and exporting natural gas to Israel at low prices.
Some Egyptians fear Mubarak, 83, will receive a lighter sentence than the death penalty.
The Youth Shadow Government said Tuesday it will collect signatures for its petition both online and in person.
They are calling their campaign "I support the death penalty for Hosni Mubarak."