CBS News Correspondent Bob McNamara reports McGinn had exhausted all appeals and was to be executed on June 1 for the 1993 rape and ax murder of his 12-year-old stepdaughter.
But 18 minutes before the sentence was to be carried out, Bush granted the man a 30-day reprieve to allow time for DNA testing.
Those tests are now complete and Federal Bureau of Investigation sources say they point to McGinn or a maternal relative as the source of a pubic hair found inside the body of Stephanie Flanary.
USA Today, which broke the story, noted that no member of McGinn's family has been linked to the case.
Flanary's stepmother was pleased with the news.
"It's a good step in the right direction but we would much rather have facts. We do think it's in the right direction," said Livia Flanary.
Although DNA evidence was used by prosecutors to help convict McGinn in 1995, the 43-year-old death row inmate and his attorneys wanted more sophisticated testing they hoped would exonerate him.
They argued that new DNA tests proving the hair was not McGinn's would show he did not rape the girl.
USA Today said the new McGinn tests were completed last month and filed with Stephen Ellis, a judge in Brown County, Texas, where the crime occurred.
Results of a second set of DNA tests on semen left at the crime scene, which could still exonerate McGinn, remain under a judge's gag order, but reports indicate it is doubtful those results will clear McGinn either.
Ellis is scheduled to rule on the impact of the tests sometime after Aug. 15, at which time he may set a new execution date.
USA Today said an FBI spokesman delined to comment on the case.
Bush had a reserved reaction.
"I know nothing more than rumors. At some point in time we will find out the truth," he said.
The way Bush has dealt with the death penalty in his two terms as Texas governor has been a campaign issue as he awaits formal nomination next month as the Republican candidate for president.
The governor was widely condemned last month for failing to grant a reprieve to convicted murderer Gary Graham, whose guilt had been questioned by two witnesses and whose trial counsel showed signs of incompetence.
But the unprecedented reprieve Bush granted McGinn is expected to help the candidate if DNA confirms McGinn's guilt, because Bush has long contended he has full confidence that the 136 people executed on his watch were all guilty.
"The real problem would have been if the evidence had come out the other way, throwing into question the whole Texas death penalty system and his administration of it," said Cal Jillson, chairman of the political science department at Southern Methodist University in Dallas.
Gary Graham's June execution sparked a firestorm of controversy.
But the resolution of the McGinn case will not end questions over the use of DNA in capital cases in Texas. Another death row inmate, 38-year-old Henry Watkins Skinner, who was convicted of three 1993 murders, has been denied DNA testing.
In an interview with CBS News Maurie Levin, one of McGinn's lawyers, discussed the importance of DNA testing, regardless of whether the results help or hurt the defendant.
"Whatever happens in the McGinn case, what is important for people to take away is the need for standardized procedures to permit DNA testing in all cases," said Levin.
She points out that states have the technology now to make sure they are executing the right person.
"You can't generalize the results of one single case to every single case. The point is, we have to be sure," Levin said.
But Steve Flanary, the victim's father, said, "It's wrong to drag the victims through it. I'd like to know how Gore would've reacted. Since this is political I'd like to know how Gore would have reacted to this case.