But Lee has consistent support among conservative and libertarian activists, including some Republicans who worry about a nominee not upholding their principles and who say the Utah senator could bring more certainty.
The President teased his potential pick on social media before the Fourth of July holiday, saying "I interviewed 4 very impressive people yesterday. He clerked for two federal judges, including Supreme Court Justice Sam Alito", Cruz wrote. "Mike Lee. I think he would be extraordinary", Cruz told Fox News on the day of Kennedy's retirement.
Kennedy, who has served on the Supreme Court since 1988, has always been recognized as the institution's lone swing vote preventing either the conservative or liberal factions from becoming ascendant.
"Indeed, numerous worst liberal judicial activists, including William Brennan, John Paul Stevens, and Harry Blackmun - the author of the Roe v. Wade decision that legalized abortion nationwide - were appointed by Republican presidents". Mike Lee, R-Utah, who is not said to be under serious consideration by the White House but is the only lawmaker Trump has considered for the position.
Citing unnamed sources, Fox News said the president has met with the six candidates, with some being interviewed a few times.
The senator's office confirmed that he spoke with President Trump on Monday - and while the White House made reference to the call, further details weren't disclosed by the administration.
The senator stated Lee was the only candidate being considered by the president who would be a "sure thing" and was likely to fire up "conservative voters" ahead of the midterm elections.
Sen. Richard Blumenthal, D-Conn., said Sunday on the CBS program "Face the Nation" that Trump doesn't need to ask where nominees stand on overturning Roe v. Wade because his list has already been screened on that issue by the Federalist Society.
The fact that Lee is a Senator means that he could, possibly, be voting on his own confirmation as a Justice, if he were to become the nominee. Schumer, the person said, pressed the president to name Garland to succeed Kennedy, arguing doing so would help unite the country.
Kavanaugh, 53, and Kethledge, 51, are both former law clerks to Kennedy, as was Neil Gorsuch, Trump's first Supreme Court nominee, who succeeded Antonin Scalia. He cited former justices William Brennan, John Paul Stevens and Harry Blackmun, who authored the Roe v. Wade decision that established a woman's right to abortion.
"Beyond the qualifications, what really matters is, does this nominee fit a central casting image for a Supreme Court nominee, as well as his or her spouse", a Republican close to the White House told Politico.