"There is more to gain from freeing the two prisoners than killing the hostage," Abu Sayyaf said in a Thursday statement.
One day earlier, the Jordanian army confirmed that ISIL militants had shot down one of its warplanes in Syria and captured the pilot, identified as Moaz al-Kasasba.
Abu Sayyaf said he had "heard" that the ISIL was considering a prisoner swap in which al-Kasasba would be traded for Sagda al-Rishawi and Ziad al-Karbouli, who were convicted of terrorism-related charges in Jordan in 2005 and 2006 respectively.
Abu Sayyaf slammed Jordan for taking part in an ongoing U.S.-led campaign against the ISIL.
"Nothing will come of this battle [against ISIL] except casualties," he said. "ISIL has never targeted Jordan's borders or threatened to attack Jordan."
Al-Rishawi, who hails from Iraq, was sentenced to death in Jordan in 2005 following a spate of bombings that targeted three hotels in Amman that left dozens dead or injured, including foreign nationals.
Al-Karbouli, who was also formally sentenced to death, was convicted in 2006 for joining an "outlawed group" and plotting "terrorist attacks."
The ISIL confirmed earlier that it had managed to shoot down a military jet and capture the Jordanian pilot in the Syrian city of Raqqa.
Photos allegedly showing the captured pilot were shared on social media by pro-ISIL users, although their authenticity could not be independently verified.
The photos show a pilot apparently in good health.
ISIL-held Raqqa has been targeted several times by the U.S.-led anti-ISIL coalition.
Over the summer, the militant group captured large swathes of land in Iraq and Syria, declaring the captured territories to be an "Islamic caliphate" and killing thousands of people – and displacing millions more – in the process.
The U.S.-led coalition has carried out numerous airstrikes against ISIL targets in both Iraq and Syria since the group captured the northern Iraqi city of Mosul in June.