While Trump hasn't named any specific contenders yet, it's widely believed that he is choosing between four federal appeals court judges-Brett Kavanaugh, Raymond Kethledge, Amy Coney Barrett, and Thomas Hardiman.
The announcement came hours before Trump is due on Monday night to name his pick to replace Justice Anthony Kennedy, marking an official start to a massive debate in the Senate on the future of the Supreme Court.
Trump spoke to reporters Sunday afternoon as he concluded a weekend in New Jersey spent deliberating his decision at his private golf club.
White House spokesman Raj Shah said in a statement that Kyl "has agreed to serve as the Sherpa for the President's nominee to the Supreme Court". Trump has been struggling with Kavanaugh's association with the Bush family, however, an issue the Post calls the "Bush factor".
And Democrats can not even be sure of holding their line against Trump's pick.
Cornyn spoke shortly after Senate Democratic Leader Chuck Schumer said any of Trump's likely nominees poses a threat to the Affordable Care Act and a woman's right to have an abortion.
Trump has previously said he wanted "pro-life" justices opposed to abortion rights. And they were the only Democratic senators who supported Trump's first Supreme Court nominee, Neil Gorsuch, last year.
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It is not exaggeration to say that the prospect of tipping the balance of the Supreme Court was the reason why evangelical conservative voters, who had plenty of reason to worry about Trump's character, chose to stick with him during the 2016 election campaign. "But we're very close to making a decision". Unlike the current Justices, Hardiman doesn't have an Ivy League law degree - a fact Trump said on the 2016 presidential campaign trail was a positive factor for a potential nominee if he became President. In recent days, he expressed renewed interest in Hardiman, the runner-up when Trump nominated Gorsuch, said two people with knowledge of his thinking. But her brief time on the bench has raised questions about her experience.
Hardiman, 53, a judge on the federal court in Philadelphia, is less known in terms of his legal philosophy, but has working-class roots that could make him attractive to the American public.
Kethledge, a former Kennedy law clerk, has sat on the 6th Circuit since 2008 after being nominated by George W. Bush. And she recently made it through the confirmation process, with the Senate approving her nomination to be an appeals court judge in October.
Here's why: Kavanaugh's court rarely issues opinions on Monday.
Raymond Kethledge of MI, who also has many high-profile supporters who see him as a positive alternative to the other three. Ten Democrats from states that Trump won are up for re-election this fall and will be under tremendous pressure to back the president's nominee.
If the nomination is delayed and Democrats capture an extra seat in the Senate in November elections, Trump could be forced to compromise with liberals in order to fill Kennedy's seat.
One Democrat from a red state, Alabama's Doug Jones, told CNN's "State of the Union" Sunday that he did not rule out voting for Trump's nominee. "I don't think my role is to rubber stamp for the President, but it's also not an automatic knee-jerk no, either". Top Senate Democrat Chuck Schumer on Monday said Trump's nominee should be obligated to make his or her views clear on matters like the Roe ruling.