Shoko Asahara's lawyers had attempted to overturn a High Court decision to reject their appeal against his death sentence.
Their petition has been dismissed, leaving them few other options to prevent Asahara being put to death.
Twelve people died and more than 5,500 others were hurt in the sarin attack.
The Aum Shinrikyo cult was found to be responsible.
After a trial lasting many years, the cult's leader, Shoko Asahara, was found guilty for his role in the attack and sentenced to death.
Asahara never gave evidence in court. His defence team and his family say his mental state has declined after years of incarceration in solitary confinement.
They believe he has no idea what is going on.
In March, his lawyers missed a deadline to file an appeal on their client's behalf because they said it was impossible to communicate with him.
The High Court rejected their explanations for the delay. A court-appointed doctor believes Asahara does understand what is happening.
The defence filed an objection. That objection has now been rejected.
The defence now has five days to attempt to get the Supreme Court to hear the case.
If those efforts are rejected, Asahara could be put to death without delay.