Yukawa, 42, was seized by militants in August after going to Syria to launch a security company. Goto, 47, went into Syria in late October seeking to secure Yukawa's release, according to friends and business associates.
The new recording, released on YouTube late on Saturday before being deleted, showed an image of a gaunt Goto in an orange t-shirt with audio of what appeared to be him making a statement in English.
"I would like to stress how easy it is to save my life," the recording says.
"You bring them their sister from the Jordanian regime, and I will be released immediately. Me for her."
Al-Rishawi was arrested shortly after she failed to blow herself up in one of three deadly hotel bombings that hit the Jordanian capital in 2005.
Japan paid $6 million to Japanese Red Army hijackers after a 1977 kidnapping, but in recent years has moved toward the U.S. government's hard line against paying ransoms.
Japan's pacifist constitution also rules out any military response. A briefing paper prepared for Abe's office on Friday and reviewed by Reuters said Japan would not have the legal authority to strike the Islamic State even after proposed legislation loosening military restrictions that the prime minister is seeking to pass later this year.
Abe told NHK that Japan did not intend to join the U.S.-led military operation against Islamic State, but wanted to continue to provide humanitarian aid. The decision by Abe to give aid specifically to countries contending with Islamic State has raised some eyebrows.
"I think it's unavoidable if they (Islamic State) took this as support for their enemies and view Japan as an enemy," Ichiro Ozawa, leader of the small opposition People's Life Party, told NHK, adding the government appeared not to know how to respond.
The Islamic State has executed five British and American aid workers and journalists in recent months. Yukawa's capture by Islamic State fighters outside Aleppo in August was the first time a Japanese citizen has been held by the group.