Lindsey Graham, R-S.C., said that while Cornyn would typically be a "superb choice" to lead the agency under normal circumstances, "these are not normal circumstances". But they would need to court at least three Republicans to be successful.
Cornyn was in serious contention to replace ousted Federal Bureau of Investigation director James Comey, setting off a scramble and speculation around the state over who might succeed him in the U.S. Senate.
Senate Democrats are demanding that Deputy Attorney General Rod Rosenstein name a special prosecutor for an investigation into Russia's influence in the 2016 USA presidential elections before moving forward with Comey's replacement.
When asked whether Sessions involvement with the search raised any concerns - since Sessions had recused himself from the Russian Federation investigation - Cornyn said he did not see any conflict.
He took the opposite position during the 2016 election, joining fellow Republicans in asking for a special prosecutor to investigate Clinton's use of a private email server for government business. Comey, for instance, was a former United States attorney in Manhattan before being appointed deputy attorney general by George W. Bush.
The opening was created when Mr. Trump fired James Comey. "The FBI's about more than just one investigation".More news: Answers from Yates, more questions for White House
More news: Number of Australian businesses hit by cyber attack rises
More news: Turkey Calls for Ouster of US Envoy for Anti-ISIL Coalition
If he had been chosen, it would have created political upheaval in Texas with both parties fighting for his Senate seat. He's active on Twitter, where he often personally posts articles critical of Democrats.
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.
Others being considered for the role include Alice Fisher, a white-collar defence lawyer who previously led the Justice Department's criminal division, and Michael Garcia, a judge on the New York State Court of Appeals who previously served as the USA attorney for the Southern District of New York.
"I think that for Cornyn you've got to remember some people see him as a partisan figure because he's in leadership, but he has much more of a judicial manner", said Texas political consultant Matt Mackowiak. He said the US Department of Justice was still interviewing candidates. Trump also is looking at former Rep.
"Where is the distinguished Republican whip?" a reporter asked McConnell.
When asked on "Face the Nation" last month if it was appropriate for Trump to comment on the ongoing Russian Federation investigation, Cornyn said "sometimes I think this is a distraction from what we should be doing".