The White House Counsel's Office, headed by Don McGahn, will supervise the process.
Raj Shah will take leave from his role in the press office to work full time on communications with Congress about the nominaton. "The Department of Justice is fully engaged to support the nomination and confirmation efforts".
"There is keen interest in those two, but there are at least a couple of others who are still very seriously in the mix", said the person.
Trump has said he would announce his nominee on July 9 and has reportedly been busy reading the writings and background material of some of the prospective candidates. In an unusually direct appeal to voters, he said that to block "an ideological nominee", people should "tell your senators" to oppose anyone from Trump's list.
Kennedy's retirement last week has also sparked debate about the upcoming midterms and whether the vacant seat will help boost Republican turnout.
And while several Republican-led states have passed aggressive regulations and would likely be emboldened by a more conservative court, there's no guarantee how the future Supreme Court will rule.
One of the candidates Trump spoke to is Utah Sen.
The president has a list of 25 candidates, which he said he has narrowed down to five, including two women.
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Three Democrats, all from states Trump won in 2016, supported Trump's prior pick, Justice Neil Gorsuch - Heidi Heitkamp of North Dakota, Joe Manchin of West Virginia and Joe Donnelly of Indiana.
Abortion rights advocates are using the #HangersForCollins and tweeting images of coat hangers a message aimed at Maine Republican Sen. John McCain not voting in the Senate due to his serious cancer, Republicans hold the barest of margins in what is now consequently a 50-49 Senate.
On Sunday, Leonard Leo, an outside adviser to Trump on judicial nominations, said he expected Trump to select a nominee who is mindful of precedent but who is also more "originalist and textualist".
With Kennedy gone, many people are predicting that Roe v. Wade could be overturned. Collins voted in favor of Gorsuch, a conservative, because he is a believer in relying on precedent and authored a book on the subject.
Nelson opposed Gorsuch past year, but he remained on the fence almost two months before making that decision. But Lee is not believed to be on Trump's short list.
Democrats, meanwhile, have begun taking aim at members of Trump's Supreme Court shortlist by name.
Trump has also been appointing young conservative judges to the lower courts at a rapid pace.
This was made known by the First Deputy Speaker, Joseph Osei Owusu on the Floor of the House today, Tuesday [July 3, 2018], following a statement the Presidency sent to Parliament. In 2006, Bush appointed Kavanaugh to the U.S. court of appeals for the DC circuit.