The video has a still picture of Mr Goto holding another picture, which seems to be of an executed Mr Yukawa.
Japan is verifying a video posted on jihadi websites claiming terror group Islamic State has killed one of two of its nationals held hostage, after a 72-hour deadline elapsed on Friday.
The video was posted on Saturday night.
Japanese Prime Minister Shinzo Abe called the video claim an 'outrageous and unforgivable act'.
His Foreign Minister Fumio Kishida, when asked if the news was confirmed, told the media, "We are aware that there is a variety of information. I will speak after confirming."
The Islamic State had released the first video on Tuesday, threatening to kill the two Japanese nationals Kenji Goto and Haruna Yukawa unless Tokyo paid a ransom of $200 million within 72 hours. That amount is the exact equivalent of what Japan had pledged to countries in the region fighting the Islamic State.
Japan has subsequently clarified that the aid was meant for refugees.
The video posted today starts with a message implying it has already been sent to the family of the hostage pictured alive and the Japanese government. It then has a still picture of Mr Goto holding another picture, which seems to be of an executed Mr Yukawa.
The video has an apparent message from Mr Goto.
He says, "(Prime Minister) Abe, you killed Haruna. You did not take the threats of my captors seriously and you did not act within the 72 hours."
With a noticeable quiver in his voice, he adds, "Rinko, my beloved wife, I love you, and I miss my two daughters. Please don't let Abe do the same for my case. Don't give up."
Mr Goto says the terrorist group no longer wants the $200 million dollars as ransom.
He says the Islamic State wants the release of an alleged failed female suicide bomber -- Sajeda al-Rishawi - who was held in Jordan. She had confessed to being part of an Al Qaeda team in Iraq that killed 57 people in an attack on the Radisson Hotel in Jordan in 2005.
While Japan has said it is verifying the authenticity of the video, the US National Security Council has also issued a statement, saying it is 'working to establish the authenticity' of the new video claim.
There have been doubts raised by experts on both videos: the video does not have the usual IS trademarks (its media wing logo, the IS flag or the usual high definition, slickly produced videos).
Mr Goto, an experienced conflict-zone reporter, made the journey to Syria sometime in October 2014. Mr Yukawa is believed to be a private military contractor who travelled to the area in early 2014 to gain experience to set up a security company. He was reportedly captured in August.
Mr Goto's mother, Junko Ishodo, had issued an impassioned plea on the day the deadline expired, saying, "Time is running out. Please save my son's life. He is not an enemy of the Islamic State."
She had also expressed anger that he had left for Syria two weeks after his son was born to try and rescue Mr Yukawa.
Japan constitutionally is not allowed to use its military to help in such situations. The US and the United Kingdom follow a strict no-negotiation policy and are reported to have been pushing Tokyo not to give in to the financial demands.
Five American and British journalists and aid workers have been killed since August 2014 in claimed IS video executions, while dozens of European hostages have been freed after reported ransom payments running in to hundreds of millions of dollars. But if the latest Islamic State video is genuine, it provides another chilling reminder of the lengths the terrorist group is willing to go to.