The video, which Weeks said was made on Jan. 1, showed the two men, both bearded, asking their families to put pressure on the U.S. government to help secure their release.
Addressing President-elect Donald Trump, who is due to take office on Jan. 20, Weeks said the Taliban had asked for prisoners held at Bagram air field and at Pul-e-Charkhi prison on the outskirts of Kabul to be exchanged for them.
"They are being held there illegally and the Taliban has asked for them to be released in our exchange. If they are not exchanged for us then we will be killed," he said.
"Donald Trump sir, please, I ask you, please, this is in your hands, I ask you please to negotiate with the Taliban. If you do not negotiate with them, we will be killed."
In September, the Pentagon said U.S. forces mounted a raid to try to rescue two civilian hostages but the men were not at the location targeted.
Kidnapping has been a major problem in Afghanistan for many years. Most victims are Afghans and many kidnappers are criminal gangs seeking ransom money but a number of foreigners have also been abducted for political ends.
Last year, the Taliban released a video showing a U.S. hostage and her Canadian husband abducted in 2012 asking their governments to pressure the Kabul government not to execute Taliban prisoners.