Rick Perry, the frontrunner to become the Republican candidate in next year's presidential election, has just hours left to prevent a man being put to death in Texas in a case in which the jury was told the prisoner was a danger to the public – and should therefore be executed – because he was black.

Duane Buck is one of four men scheduled to die by lethal injection in Texas, where Perry is governor, over the next eight days – an exceptional rate even in this execution-happy state. At Buck's sentencing hearing, the jury that set his punishment was informed by a psychologist that black people had a higher rate of violent behaviour, a statement used by the prosecution as its key argument against giving him an alternative penalty of life imprisonment.

On Tuesday night, another hotly contested case is scheduled to reach its climax with the execution of Steven Woods, who was sentenced to death for a double murder, even though an alleged accomplice later confessed to having pulled the trigger.

How Perry reacts to the demands for commutation and clemency in these two highly controversial cases will give an indication of how he proposes to deal with the death penalty issue, which has welled up in the presidential race for the first time. Perry, as governor of Texas, has presided over more executions than any other US official in modern times.

Photograph: Robyn Beck/AFP/Getty

Perry was questioned about his enthusiasm for the death penalty at a televised Republican debate last week. When the TV moderator put it to him that his state had executed 234 prisoners since he became governor in 2000, the Republican studio audience cheered.

Perry said he had never lost any sleep worrying that some of those individuals might have been innocent. "I've never struggled with that at all," he said.

When asked how he felt about the audience applauding so many deaths, he replied: "I think Americans understand justice."

Lawyers for both Buck and Woods are engaged in frenzied last-minute lobbying to Perry and to the courts to try to put off the executions. If their efforts fail, Woods's execution on Tuesday night will be followed by Buck's on Wednesday night.

Responsibility for the execution going ahead, despite the controversy over the racially-tinged testimony, is now falling squarely on the shoulders of Perry. The Texas Board of Pardons and Paroles has cleared away one of the last impediments to the ultimate penalty going ahead by refusing to recommend that Buck should be granted clemency.

That leaves Perry, who has power to issue a 30-day reprieve but who has very rarely done so.

Buck, 48, shot and killed Debra Gardner, his former girlfriend, and a friend of hers, Kenneth Butler, in a drunken explosion of jealousy in July 1995. His guilt is not in dispute, but the testimony presented to the jury at his sentencing is.

At the hearing, a psychologist, Dr Walter Quijano, was called by the defence and testified that he did not believe Buck would be a future danger as the murders had been a one-off crime of passion. But under cross-examination, the prosecution pressed him about Buck's ethnicity as an African-American.

"You have determined that the … race factor, black, increases the future dangerousness for various complicated reasons. Is that correct?" the prosecution asked.

"Yes," replied Quijano.

The prosecution later exhorted the jury to make their decision on the basis of Quijano's testimony. The jury found that Buck did pose a future danger of violence, and put him on death row.

In 2000, the then attorney general in Texas, John Cornyn, admitted that the racial testimony of Quijano had wrongfully been allowed to prejudice sentencing in seven separate cases. Six of those cases were reheard as a result, but, in a legal oversight, Buck's never was.

Buck's lawyer, Katherine Black, is petitioning Perry to commute his execution to allow resentencing. "This case violates the US constitution and undermines our moral values. A person has a right to be sentenced based not on the colour of their skin," the petition reads.

Further pressure has been brought to bear on Perry by a senior Texas lawyer who acted as prosecutor in Buck's original trial. Linda Geffin has written to Perry calling on him to delay the execution. "It is inappropriate to allow race to be considered as a factor in our criminal justice system," she wrote.

Steven Woods, 31, who will die barring a last-minute stay of execution, was one of two men accused of murdering Ronald Whitehead and Bethena Brosz in a drugs turf war in May 2001.

Woods was brought to trial in August the following year. The prosecution alleged that he had planned and carried out the shootings, and he was convicted and sentenced to death.

Three months later, his alleged accomplice, Marcus Rhodes, who had cut a deal with prosecutors, was given a life sentence, despite having confessed that he had personally carried out the shootings. Rhodes was given life imprisonment, while Woods remained on death row.

Amnesty International has issued an urgent action alert, accusing Texas of treating Woods unfairly in a case "where one defendant receives a death sentence and another who pled guilty to personally shooting the two victims receives a life sentence".

Mary O'Grady, a specialist in death row based in Austin, said that under the so-called "law of parties" in Texas, death penalties can be inflicted even on those who did not pull the trigger. Being present at a murder, knowing that an accomplice intended to kill, is sufficient.

"A lot of people with no blood on their own hands get executed in Texas," O'Grady said.

The prospects of Perry granting clemency for Woods are not great. The governor has only once in 11 years shown clemency to a death row inmate unless forced to do so by the courts.

"When it comes to death row, Perry is completely unfeeling and unemotional," said Ray Hill, who runs the Execution Watch website and radio show in Texas.

"It never strikes him that he should value the lives of those who are accused, even wrongfully."

Next week two further executions are scheduled, of Cleve Foster on Tuesday and Lawrence Brewer on Wednesday.