Scott Panetti, who has had schizophrenia for three decades, has won support for his case of groups such as Mental Health America, psychiatrists, former judges and prosecutors, evangelical Christians.
The European Union urged Texas Governor Rick Perry to grant Panetti clemency.
"The execution of persons suffering from a mental disorder is contrary to widely accepted human rights norms and is in contradiction to the minimum standards of human rights set forth in several international human rights instruments," the bloc wrote in its letter.
"As an obviously severely mentally ill man with schizophrenia, Mr Panetti should never have been allowed to represent himself in his death penalty case," his attorney Kathryn Kase said.
He "should not have been allowed to reject a plea deal that would have saved his life. Now, Mr Panetti must not be executed without a competency hearing.
"This is the last chance to prevent an injustice from turning into an immoral tragedy."
Though individual US states choose whether they will implement the death penalty, in 1986 the US Supreme Court barred execution of the mentally ill as cruel and unusual punishment.
