Scott Panetti, 56, is slated for execution on Wednesday for the fatal shootings of his in-laws in 1992.
High court justices in 2002 prohibited the execution of the mentally impaired, but have allowed it for mentally ill inmates with a rational understanding.
Panetti's lawyers argue the mentally ill should also be exempt under law.
A number of conservatives leaders have also joined the fight to save his life, writing a letter asking Texas Governor Rick Perry to commute the death sentence to life in prison.
"As conservatives, we must be on guard that such an extraordinary government sanction not be used against a person who is mentally incapable of rational thought," according to the letter.
"It would be immoral for the government to take this man's life."
The Texas inmate was severely mentally ill "before, during and after the crime for which he has been sentenced to death", lawyers Gregory Wiercioch and Kathryn Kase told the Supreme Court on Monday.Panetti was diagnosed with schizophrenia in 1978 and hospitalised more than a dozen times before killing Joe and Amanda Alvarado.
"Imposition of the death penalty on people with severe mental illness, as with people with intellectual disability, does not serve the two goals of deterrence and retribution because of their reduced moral culpability," his lawyers told the court.
The Supreme Court added a provision mandating that an inmate have a rational understanding of why he was being put to death in 2007 under a previous appeal from Panetti.
His case has gone to the high court for review at least five times since his 1995 conviction, records show.
A separate appeal for the inmate - requesting a delay in execution to evaluate his mental competency - is currently before a federal appeals court.
Ellen Stewart-Klein, an assistant Texas attorney general, told that appeals court Panetti's medical records "strongly indicate rational awareness of his impending execution and the reason for it".
"Panetti's mental status has at best been severely exaggerated by his counsel," she added.
