Conservative Senator Mike Lee (R-Utah) said President Trump should nominate Judge Merrick Garland to replace James Comey as FBI director.
Lee's suggestion was initially met by mixed reaction, because Garland was President Obama's pick to succeed Justice Antonin Scalia on the Supreme Court.
The Republican-controlled Senate denied Garland a hearing pending the outcome of the 2016 election, and President Trump later nominated Justice Neil Gorsuch.
Garland was nominated by President Clinton to the District of Columbia Circuit Court of Appeals, where he currently sits.
Lee noted that Garland presided over several high-profile cases, including that of Theodore Kaczynski - also known as the "Unabomber" - and 1995 Oklahoma City bomber Timothy McVeigh.
Lee said Garland's political popularity among Democrats and his judicial record make him a perfect candidate to receive bipartisan support in the Senate.
"I'm absolutely serious about it," Lee told Chris Wallace on "Fox News Sunday."
He said it is important in the current political climate that whoever Trump nominates be someone untainted by partisanship.
Nearly a dozen other names have been floated for Comey's replacement, including 9/11-era NYPD Commissioner Ray Kelly, FBI Richmond, Va. chief Adam Lee, former Rep. Mike Rogers (R-Mich.), and interim FBI Director Andrew McCabe.