"I have not been asked for specific documentation and have not had a specific conversation with any of his staff about being vice president," Fallin told CNN's Jake Tapper in an interview that aired Sunday on "State of the Union." She added, however, that "it's certainly an honor to be mentioned with other people across our nation."
In May, Trump campaign chairman Paul Manafort told The Huffington Post that the billionaire was unlikely to pick a woman or a minority so as to avoid being seen as "pandering."
Asked whether the GOP would be missing an opportunity were Trump to select a white man to be his vice president, Fallin responded that she hopes the campaign will focus on a candidate's qualifications, not their gender.
Fallin's remarks come one week after another woman thought to be a potential Trump running mate, Iowa Sen. Joni Ernst, indicated that she was not interested in the position
Two other women occasionally mentioned in connection with Trump's VP spot are former Arizona Gov. Jan Brewer and Tennessee Rep. Marsha Blackburn, who told CNN last week that she "really can't say" if she was being vetted but that she would be an "unlikely candidate."
On Saturday, a source familiar with Trump's vetting process told CNN that former House Speaker Newt Gingrich is "actively lobbying"
for the job. Indiana Gov. Mike Pence is also being vetted, and retired Army Lt. Gen. Michael Flynn and New Jersey Gov. Chris Christie are still in the mix, the source added.
Running as a 'racial healer'
Fallin, speaking after a week of heightened racial tension following shootings in Louisiana, Minnesota and Dallas, also defended the billionaire's controversial campaigning style. Trump on Friday mourned the deaths of Alton Sterling and Philando Castile, who were killed this week at the hands of police officers. But he called the murders of five officers in Dallas "an attack on our country" and said racial divisions in the country
have "gotten worse, not better."
"I think he's trying to campaign as a racial healer," Fallin said. "I think that has been part of his message, if you watched what he said this week. You know, he talked about how devastating this was for Dallas, how we have to respect our law enforcement and we need to pray for those who were killed and injured."
"I think that is his intent. I trust him with his words," she added.
Fallin called the police ambush in Dallas a "horrendous, terrible tragedy," and said, "that's not what America's about ... the enemy is not the enemy within ourselves. Our enemy is outside of our nation." The governor said she saw "great people" at the protests "expressing their concerns" and that we "need to listen to those voices" who are speaking out against police brutality.
"It's important as a nation that we have a dialogue" she said.