Jose Ernesto Medellin, convicted of the 1993 rape and murder of two teenagers, is among 51 Mexicans on death row. But their government has complained they were not informed of their right to consular access and assistance during trial, a right under the Vienna Convention.
Medellin's request for a reprieve was unanimously turned down on Monday by the Texas Board of Pardons and Paroles. His fate lies in the hands of Texas Governor Rick Perry, who is expected to make an announcement on the issue, his spokeswoman said.
Medellin's lawyers have also appealed to the US Supreme Court for a stay of execution.
The ICJ in 2004 ordered US officials to review the sentences for the Mexicans on death row.
On July 16 this year, the ICJ instructed the US authorities to do everything it could to stay the imminent execution of five Mexicans on death row, including Medellin, granting an urgent request by Mexico.
Since the 2004 ruling, some US states have agreed to review their death row cases at President George W. Bush's request.
But Texas has refused, arguing that its state courts, which decided the Medellin case, are not bound by the ICJ treaty. And their position was supported by a March US Supreme Court ruling.
That left the federal government in Washington with no legal tools to force Texas to put off the execution.
Medellin's lawyers are still hoping the US Supreme Court will issue a stay of execution that would give the US Congress time to pass a new law that can force individual states such as Texas to abide by ICJ decisions.
The US House of Representatives took up such a bill after the ICJ's July 16 ruling.
Alternately, Medellin's lawyers are seeking at least an eight-month reprieve so that the Texas legislature -- in recess until January -- can approve a law allowing state or federal court hearings for their client and other Mexicans on death row.
"Should Texas execute Mr Medellin before Congress has a reasonable opportunity to convert the (ICJ) judgment into a justifiable federal right, the State of Texas will forever deprive Mr Medellin of his constitutionally protected right not to be deprived of his life without a due process of law," his lawyers told the Supreme Court in their motion this week.
They asked the high court "to ensure that its judgment has its intended effect of guiding the political branches to a constitutionally permissible method of complying with the nation's treaty obligations."
Congress however is in recess until September. Any such measure would have to be approved by both the House and the Senate, then signed into law by the president.
Human Rights Watch, in a weekend statement, said that "executing Jose Medellin in violation of an order by the World Court would be a major step backward for the rule of law."
"If the United States ignores its legal obligations in this case, it will be tough to argue that other countries should respect the rights of US citizens," it added.
The Texas board on Monday also rejected a request by Medellin's lawyers' to commute his sentence to life in prison.
Medellin was sentenced to die for the rape and murder of two teenage girls -- aged 14 and 16 respectively -- in 1993 in Houston. The girls happened upon Medellin, who was 19 at the time, as he was engaged in a street gang initiation rite.
© 2008 Agence France Presse