He says he doesn't believe the FBI investigation into any Trump-Russia ties is at risk.
"He didn't recuse from being attorney general", Cornyn told reporters.
The Department of Justice flatly denied those reports.
U.S. Rep. Henry Cuellar said in an interview last week that he has been especially "impressed" with Cornyn's comments over recent months on trade and NAFTA.
Compared with the other choices interviewed by Sessions and the Department of Justice over the weekend, Cornyn is well-known to his former senators.
Garcia, an associate judge of the New York Court of Appeals, was nominated to that court by Gov. Andrew Cuomo in January of 2016 and confirmed a month later. Whoever is selected is appointed to a 10-year term, although they can be removed by the president. Trump abruptly fired Comey, allowing Trump surrogates and even his vice president to put forth an explanation that Trump himself discarded on NBC Nightly News two days later.
Mackowiak said that whomever Trump nominates will likely be confirmed "in the 51- to 54-vote range" because of the fallout after Comey's dismissal, and that the president can not nominate a loyalist like former New York City mayor Rudy Giuliani to the post.
"We don't have press conferences", Trump said.
Cornyn is former Texas Supreme Court judge and attorney general, a senior member of the Senate Judiciary Committee, and a member of the Senate Select Committee on Intelligence.
John Cornyn, the majority whip and second-ranking Republican, was not present in the gaggle of Senate leaders after their weekly lunch Tuesday.
"Well, my recommendation to the president, who apparently said today he was not interested, I'd recommended he take a look at Judge Merrick Garland", McConnell said.
"Now more than ever the country needs a well-credentialed, independent FBI Director", Sen.
Cornyn and Gowdy likely would have faced a hard confirmation process given their partisan backgrounds. They said sudden interest was a GOP strategy to open a seat on the circuit court for a Trump administration nominee.
"We asked Senator Cornyn to instead visit with our students again at a future date in order to keep the focus on graduates and their families", the post said. "All it does is heighten the attention given to the issue".
We reached out to Senator Cornyn's office about the interview.
Cornyn had also been scheduled to deliver a commencement address on Saturday at the Texas Southern University in Houston, but university officials on Friday cancelled his appearance at the historically black college after a petition signed by hundreds of students protesting his appearance.
Cornyn's spokesman said, "Sen".
Rosenstein and acting FBI Director Andrew McCabe were invited Friday to brief the Senate Judiciary Committee about the FBI's Russian Federation investigation.
Comey, who is now a private citizen, but one who knows more than most about the Russian Federation probe, was invited to brief the Senate intelligence committee Tuesday but turned it down for now.
If chosen, Cornyn would lead the Federal Bureau of Investigation for a 10-year term.
Still, the administration has considered roughly a dozen candidates to replace Comey-a list that includes top lawmakers as well as law enforcement figures.