Edgar Tamayo Arias, 46, was convicted of the 1994 murder of a Houston police officer.
His supporters say he was denied access to his consulate when arrested, as required by an international treaty.
Mexico condemned the execution, scheduled for 7 p.m. ET in Huntsville, saying it violates an International Court of Justice ruling ordering the United States to review capital convictions of Mexican nationals.
Earlier Wednesday, the state's Board of Pardons and Paroles denied Tamayo's clemency request.
The Bush and Obama administrations had urged Texas and other states to grant Tamayo and inmates in similar situations new hearings, fearing repercussions for Americans arrested overseas.
Since 2008, Texas has executed two other Mexicans convicted of murder who raised similar claims. The Supreme Court refused to delay either one.
Tamayo's lawyers argue the consulate access violation was more than a technicality -- that Mexican officials would have ensured he had the most competent trial defense possible, if they had been able to speak with him right after his felony arrest.
Houston police officer Guy Gaddis was fatally shot after arresting Tamayo and another man for robbery.
Secretary of State John Kerry appealed to Texas Attorney General Greg Abbott this week to delay the execution.
Tamayo is one of 40 Mexican citizens awaiting lethal injection in U.S. prisons.