McConnell is concerned about the volume of the documents Kavanaugh has created in his 12 years on the U.S. Court of Appeals for the District of Columbia Circuit, as well as in his roles as White House staff secretary under President George W. Bush and assistant to Kenneth W. Starr, the independent counsel who investigated President Bill Clinton.
Mr Trump is due to make his announcement on Monday.
While final nominees will be known only when Trump makes the announcement, this would be for the second time that Thapar might not be able to make the final cut after being interviewed by Trump.
Mr. Thapar was on the shortlist of candidates after Judge Antonin Scalia died in 2016.
"Just as he did with Neil Gorsuch, the president has promised to nominate an impartial judge, a wise and seasoned jurist committed to upholding the Constitution at all costs", he writes.
McConnell was originally hopeful that Trump would select Amul Thapar, a federal appeals court judge who was previously on the bench in Kentucky, but has concluded that the president is unlikely to name him.
Talking to reporters travelling with him to Montana overseas Air Force One, Mr Trump said he has interviewed some "extraordinarily talented and brilliant" people during the process.
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The pause will give divers time to prepare gear, including masks and refilling air tanks, for the next round of rescue efforts. The boys were immediately taken to the hospital for evaluation after being trapped underground for almost two weeks.
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It's been more than a week since they released a tally of how many kids from separated families remain in government custody. Officials have said they'll comply with the judge's order , but they haven't specified what steps they'll take to do that.
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I don't know if that's two or 10, or somewhere in-between. "I think everyone needs to recognise that". While it was a painful experience, it actually brought us closer .
"We're going to do it at 9 pm in the White House", he said. This will be 6.30 am IST, 10 July.
Trump has teased details of his process in recent days, saying Thursday that he was down to four people and "of the four people, I have it down to three or two".
"If we put another two or perhaps three justices on". All but one of the 25 people on the list are members of The Federalist Society, a national organization of conservative lawyers, and there's a clear pattern to judicial nominees being fed by the group: They are young, conservative and have records of being very opposed to abortion rights.
He said he believes that any of the four candidates on the short list will dismantle Roe v. Wade and other hard-fought rights. "I've never seen a president of the United States in effect make himself a puppet of outside groups and choose from a group of right-wing fringe ideologues that are prepared on this list". "I'll give you the answer", Mr Trump said.
Trump had discussed the nomination with Vice President Mike Pence in Bedminster, New Jersey yesterday.
Ginsburg was confirmed by the senate by a whopping 96 votes to 3.