One source said that Trump favored Cornyn for the post.
The President called Cornyn to discuss the position twice, including after Cornyn passed up the job.
Cornyn, who the officials say was intrigued by the idea and initially interested, ultimately withdrew from consideration after a number of his colleagues said they believed Trump should shy away from picking a GOP politician.
Among the candidates, Trump will meet with 2000 Democratic vice presidential candidate-turned-independent Joe Lieberman, a former senator who supported Republican Sen. John McCain in the 2008 presidential election.
Trump will also meet with acting FBI director Andrew McCabe, former Oklahoma Gov. Frank Keating and Richard McFeely, a career FBI official who retired in 2014 as head of the Criminal, Cyber, Response and Services branch.
Lieberman told CNN the appointment "was not sought after or expected," but that he got a call Tuesday asking him to fly into Washington for the interview.
Keating, the governor of Oklahoma during the 1994 bombing of the federal building in Oklahoma City, left the White House with a smile on Wednesday.
"We had a good conversation," Keating told reporters. Asked if he would accept the job if offered, he said that he liked public service.
The meetings come a day after news broke that Trump allegedly asked Comey in February to drop the FBI's investigation into his former national security adviser Michael Flynn.
"I hope you can see your way clear to letting this go," Trump told Comey, according to a memo Comey wrote at the time, excerpts of which leaked to reporters on Tuesday.
The memo only added to the furor surrounding Trump's sudden and controversial firing of Comey last week, which came amid the FBI's intensifying probe into Russian interference in the 2016 campaign and Trump campaign associates' potential collusion with Moscow officials.
That firing has sparked heated questions from members of Congress in both parties who have called for a closer look into why Comey was fired and whether it was tied to the Russia investigation. Comey is expected to testify before the House Oversight Committee next Wednesday.
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.
One Senate Democratic leadership aide responded to the news by saying, "There couldn't be worse time to take the unprecedented step of handing the FBI over to a politician. That includes Sen. Lieberman."