In an announcement on state television, Mohammad al Momani said: "Jordan is ready to release prisoner Sajida al Rishawi if the Jordanian pilot Lieutenant Muath al Kasaesbeh was released and his life spared."
Islamic State has threatened to kill Mr Goto and a Jordanian pilot being held by the group "within 24 hours".
The news comes as the mother the 49-year-old journalist of made another tearful appeal to Japan's Prime Minister to help save her son.
Junko Ishido said she had begged Shinzo Abe to "Please save Kenji" and to work with the Jordanian government to secure his release.
The latest video from Islamic State features another still of the freelance journalist with an audio message of a man speaking, who claims to be Mr Goto. The voice can be heard saying: "I've been told this is my last message. Time is now running very short; it is me for her."
The warning came only hours after a Japanese envoy in the Jordanian capital, Amman, voiced hopes both Mr Goto and Jordanian pilot 1st Lt. Mu'ath al-Kaseasbeh, who's been held by the extremist group after crashing in northern Syria in December would be freed.
"I hope we can all firmly work hard and join hands to cooperate between the two countries (Japan and Jordan), in order for us to see the day when the Jordanian pilot and our Japanese national Mr. Goto can both safely return to their own countries with smiles on their faces," said Yasuhide Nakayama, a deputy foreign minister and lawmaker who was sent to Amman to coordinate efforts to save two Japanese hostages held by the Islamic State group, reports AP.
Talks between Japan's Prime Minister, Shinzo Abe and Jordanian King Abdullah II are believed to have begun on Saturday after militants said they would release the remaining Japanese journalist Kenji Goto if a prisoner called Sajida al Rishawi, an Iraqi woman sentenced to death in Jordan for involvement in a suicide bombing that killed 60 people was released from jail in Jordan.
Over the weekend, an unverified video surfaced showing a still photo of Goto, 47, holding what appears to be a photo of fellow Japanese hostage Haruna Yukawa, who IS say they beheaded.
The message retracted a demand for payment of $200 million in ransom for the two Japanese, made in an earlier online message and threatened to kill Goto unless al-Rishawi was released.
Yukawa was captured last summer, and Goto is thought to have been seized in late October after going to Syria to try to rescue him.