The move is part of a broader push by the White House to rapidly confirm a replacement for Kennedy before the court's new term begins in October.
White House press secretary Sarah Sanders also refused to name them, but did say each of the four spent about 45 minutes with the President. Trump also spoke about the increasing number of Democrats who have called for abolishing Immigration and Customs Enforcement (ICE), something Trump said he considers also considers beneficial for Republicans headed into the midterms.
Trump has said he will announce his Supreme Court nominee on July 9.
In the Oval Office on Monday, he said the "four potential justices" he met that morning were "incredible people in so many different ways".
"To allow states to be able to go back to criminalizing women's freedom to make their own healthcare decisions, I think would be devastating", Kaine said.
The public is also split on whether the Court has a partisan lean, with 31% saying it is too conservative, 19% feeling it is too liberal, and 41% believing it is just right.
The White House also announced Monday that the overall confirmation process would be led by White House counsel Donald McGahn, as it was during the process that led to the successful confirmation a year ago of Justice Neil Gorsuch.
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. That creates a unique problem for Republicans, and an unexpected opportunity for Democrats.
Republican Senator Susan Collins is being sent coat hangers in the mail by abortion rights activists keen to encourage her to vote down any Supreme Court nominees who would threaten Roe v. Wade.
Candidate Bret Kavanaugh said he considered Roe vs. Wade a "settled law" during his federal court confirmation hearing in 2006.
In the case of a 50-50 vote on the nominee, Vice President Mike Pence could cast the tie-breaking vote.
In this file photo, Supreme Court Associate Justice Neil Gorsuch, June 1, 2017, is seen during an official group portrait at the Supreme Court Building in Washington, D.C.
For now, the liberal groups have launched a campaign against all candidates on Mr.Trump's list.
While Kennedy was a conservative, he proved to be a somewhat unpredictable "swing" vote over his long career.
Kennedy was 80 when he announced his retirement.
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The president nominated 34 judges to the United States Court of Appeals, and the Senate has already confirmed 20. But she added, "If he does end up nominating a woman, she can stand on her own accomplishments".