The Holy Jihad Brigades addressed the US government in a statement issued on Wednesday, which read: "Release what you have, and we will release what we have."
"Free detained Muslims in the United States in exchange for the release of the detainees in our hands," the statement continued.
Steve Centanni, 60, an American Fox correspondent and cameraman Olaf Wiig, 36, a New Zealander, were abducted by masked gunmen in Gaza City on August 14 while working on a story.
"We will give you an offer that we will not repeat," said the group's statement.
"We will give you one chance that will not be repeated - the liberation of Muslims detained in American prisons in exchange for the detainees in our hands.
"We give you a deadline of 72 hours starting from today at noon to decide. If you satisfy our condition, we will keep our promise."
The statement did not say what would happen if the group's demands were not met.
Hostages appeal for help
A video released with the statement showed the two journalists dressed in tracksuits. The men said that they were in good health and asked their governments to do everything to secure their release.
Centanni said they had access to showers, bathrooms and food.
"Our captors are treating us well," Centanni said.
"So, just want to let you know I am here and alive and give my love to my family and friends and ask to do anything you can to try to help us get out of here," he said.
Wiig asked the New Zealand government to help free them.
"If you could apply any pressure on the local government here in Gaza and the West Bank that would be much appreciated by Steve and myself," Wiig said.
The largest armed Palestinian groups in Gaza, Fatah - a secular nationalist party - and Hamas and Islamic Jihad, both Islamist groups, have previously denied any involvement.
About 30 Palestinian journalists staged a demonstration on August 19 to call for the release of the men, carrying banners that said "No to the kidnapping of journalists. Yes to freedom of the press."
Mahmoud Abbas, the Palestinian president, has said that the release of Centanni and Wiig is a priority for his government.
Fox News is a US-based news channel owned by Rupert Murdoch's News Corporation.