Mr Trump has chosen 9pm on Monday evening, a primetime TV slot, to make his announcement.
"I think we can confirm any of the four names being mentioned", Blunt said on NBC's "Meet the Press". Among those believed to be on the short list are federal appeals court judges Brett Kavanaugh, Amy Coney Barrett, Raymond Kethledge and Amul Thapar.
The confirmation for Trump's second court pick is expected to be contentious, with the Senate now narrowly divided, 51-49, in favor of Republicans.
White House Press Secretary Sarah Sanders announced that the White House Counsel's office, led by Don McGahn, will again oversee the selection and confirmation process.
Before retiring in 2013, Kyl was a member of the Senate Judiciary Committee, which will be the first to consider the nomination.
"In choosing a new justice, I will select someone with impeccable credentials, great intellect, unbiased judgment and deep reverence for the laws and Constitution of the United States", Trump said in his weekly radio address released Friday.
His concerns mirror comments from some conservatives who view Kavanaugh as a more establishment-aligned pick on abortion and issues related to the Fourth Amendment right against unreasonable searches and seizures.
Trump has also spoken with Thomas Hardiman, who has served with Trump's sister on the 3rd U.S. Circuit Court of Appeals in Philadelphia, according to a person familiar with the conversation who also was not authorized to publicly discuss it. Some also raised concern about his approach to the Affordable Care Act.
Kavanaugh's record as a judge could hold some appeal to moderate lawmakers if Trump gives him the nod. Asked about the call, White House spokesman Raj Shah said: "Yesterday, the President spoke on the phone with Sen". Mike Lee on Monday.
Trump, who is spending the weekend at his golf resort in New Jersey, is still considering other candidates. A graduate of University of Michigan Law School, he clerked for Kennedy in the late 1990s and advised Republican Senator Spencer Abraham of Michigan.
"The court could use some perspectives that were forged in different kinds of regions of the country and different kinds of academic backgrounds", said former Michigan Sen.
Aware that judicial picks are key voting issues, Trump has stressed that he wants a justice who will be a strict constitutionalist.
Bonauto noted that a year ago, then-nominee Neil Gorsuch refused to answer questions about certain cases, such as Brown v. Board of Education, and the senators should not allow that to happen this time. Barrett has excited social conservatives since she was questioned about her Roman Catholic faith in her nomination hearings a year ago, but her brief time on the bench has raised questions.
Yet, at that time, he did not go as far as to say that US citizens should lose the right to an abortion that was legalized by the 1973 Roe v Wade ruling.
Because he often positioned himself at the fulcrum between the court's liberal and conservative wings, the 81-year-old jurist will leave the nation's highest tribunal at the end of this month as he arrived there in 1988 - surrounded by controversy over filling its so-called swing seat.
MostDemocrats and Independents support Roe v. Wade: 84 percent of Democrats and 68 percent of Independents said they agree with the landmark decision.