An American woman and her Canadian husband have made an emotional appeal to President Barack Obama to work to secure their freedom before he leaves office, after four years in captivity in Afghanistan.

Caitlan Coleman and Joshua Boyle were seized in October 2012, during a hiking holiday through Russia, Central Asia and Afghanistan.

Ms Coleman, 28 at the time, was pregnant with her first son. A second son has since been born, and the video released on Monday, dated December 3, was the first time her sons have been seen.

In the clip, Ms Coleman, from Pennsylvania, begs for an end to their "Kafkaesque nightmare."

“Please don’t become the next Jimmy Carter,” said Ms Coleman, 31, in an apparent reference to Mr Carter’s leaving the White House without securing the freedom of the Iran embassy hostages. “Just give the offenders something so they and you can save face so we can leave the region permanently.”

It was unclear if she was reading from a script, but she was critical of both the American and Canadian governments, and their captors. The family is believed to be being held by the Haqqani network, an offshoot of the Taliban that roams the borderlands between Pakistan and Afghanistan.