Mexico says implementing the death penalty would violate international treaties.
Edgar Tamayo, 46, is set for lethal injection in a Texas facility in a few hours.
He was convicted of shooting dead Houston officer Guy Gaddis, 24, who had arrested him on suspicion of robbery.
While handcuffed and in the police car, Tamayo pulled a gun that had gone unseen and shot Mr Gaddis three times in the back of the head.
He kicked open a window and ran away from the car but was arrested again a few blocks away.
Tamayo's lawyers and the Mexican government contended the case was tainted because he was not advised under an international agreement that he could get legal help from his home nation following his arrest.
In a recent statement, the Mexican Foreign Ministry said: "If Edgar Tamayo's execution were to go ahead without his trial being reviewed and his sentence reconsidered ... it would be a clear violation of the United States' international obligations."
Secretary of State John Kerry has asked Texas Attorney General Greg Abbott to delay Tamayo's punishment, saying it "could impact the way American citizens are treated in other countries".
Governor Rick Perry could still stay the execution, though he has rejected such calls in the past.
"It doesn't matter where you're from," Mr Perry's spokeswoman Lucy Nashed said.
"If you commit a despicable crime like this in Texas, you are subject to our state laws, including a fair trial by jury and the ultimate penalty."
Tamayo was in the US without legal permission and had a criminal record in California, where he had served time for robbery and was paroled, according to prison records.
Texas has executed 508 prisoners since the reinstatement of capital punishment by the Supreme Court in 1976, the most of any US state.
(c) Sky News 2014
Original headline: Texas Set For Execution Of Mexican Cop Killer
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