After Trump fired Comey last week amid the FBI's investigation, Cornyn said it was "within his authority" to do so.
The Texas Republican said in a statement that the best place for him to serve is in the Senate.
Susan Collins (R-ME) said, "He would be a great choice in normal times, but we're not living in normal times".
John Cornyn for the Federal Bureau of Investigation director position last weekend, two Republicans familiar with the process told CNN Wednesday.
The Houston Republican issued a statement saying he believes he can best serve is fellow Texans and his country by remaining in the Senate.
Responding to an email inquiry from Breitbart Texas, Senator Cornyn wrote on Friday, "I have the distinct privilege of serving 28 million Texans in the United States Senate, and that is where my focus remains".More news: Senate panel wants James Comey to testify
Trey Gowdy, a SC representative and former prosecutor known for his no-nonsense style in committee hearings, was another popular name from Capitol Hill to surface. The source declined to be named because the decision was private.
Rep. Trey Gowdy, a South Carolina Republican who oversaw the House Benghazi Committee also withdrew his name from consideration. His name was floated as a possible replacement for Director Robert Mueller in 2013.
Lieberman's selection could be controversial and while he was for a time a Democrat, his exit from the party and eventual support for McCain in 2008 has caused many Democrats to sour on their party's 2000 vice presidential nominee. After Trump won the election, the high court vacancy was filled last month by Neil Gorsuch.
That's according to two White House officials briefed on the matter who spoke on condition of anonymity to discuss internal plans.
On Monday, Cornyn, who has insisted he has not been offered the job, told CNN he interviewed this weekend for 45 minutes with Attorney General Jeff Sessions, one of his former Senate colleagues, and Rod Rosenstein, who is the deputy attorney general.
Cornyn declined to say Monday evening whether the Russian Federation investigation came up during his interview, which Attorney General Jeff Sessions attended. In addition to members of Congress, the list includes current and former FBI and Justice Department leaders and federal judges. Gen. Alice Fisher, New York appeals court Judge Michael Garcia, FBI special agent Adam Lee and U.S. District Judge Henry Hudson of Virginia.