Cornyn said in a statement that he had informed the White House that "the best way I can serve is continuing to fight for a conservative agenda in the US Senate".
Trey Gowdy was also in the running to replace Comey. But Cornyn declined to say whether the Russian Federation probe or Comey's firing were discussed.
NPR reported Tuesday that Judge Merrick Garland, a novelty candidate for the post suggested by Utah senator Mike Lee and endorsed by Senate majority leader Mitch McConnell, has no intention to leave the bench.
Asked if he would accept the job if offered, he said that he liked public service.
"He didn't recuse from being attorney general", Cornyn told reporters.
Cornyn, who has served as a senator since 2002, says while he won't be filling the position, he believes the country "needs a well-credentialed, independent FBI Director".
The administration has interviewed at least eight candidates to replace Comey, of more than a dozen being considered.
Graham was referring, obviously, to Trump's clumsy decision to fire Comey, which, as the president told NBC's Lester Holt, was accompanied by the thought that "you know, this Russian Federation thing with Trump and Russian Federation is a made-up story".More news: Ohio unemployment rate inches down to 5 percent for April
A source familiar with Cornyn's thinking told Fox News he felt obligated to consider the opportunity of leading the Federal Bureau of Investigation out of a desire to restore stability - and because his close friend, Attorney General Jeff Sessions, asked him.
While Cornyn is one of the more popular members of the Senate GOP Conference, a number of his colleagues are skeptical that he would be the right choice for the job, anxious that such a pick would be viewed as too political at a time when many critics are questioning whether Comey was sacked to slow-walk the FBI's investigation into alleged Russian coordination with the Trump campaign during last year's elections.
Democrats are expected to raise fierce concerns about any nominee seen as a partisan defender of Mr. Trump and Senate Republican leaders are hoping to avoid such a conflict.
"We are moving quickly and expeditiously to pick an interim and a permanent replacement, and we're doing our due diligence -we're not going to cut any corners", a source told Fox News.
"He has an extensive background in the criminal justice field", Rottinghaus says.
May 3, 2017: Testifying before the Senate Judiciary Committee, Comey insists he was consistent in his handling of the separate investigations into Clinton and Trump.
A look at key moments in Comey's tenure, the lead-up to Trump's decision to fire him and developments since then.