Collins says a nominee who demonstrated "hostility" to the Roe decision "would not be acceptable to me".
The White House is focusing on five to seven potential candidates to fill the vacancy of retiring Justice Anthony Kennedy, a swing vote on the court.
With Trump committed to picking from a list of 25 potential nominees that he compiled with guidance from conservatives, Schumer said any of them would be "virtually certain" to favor overturning Roe v. Wade, the 1973 case that affirmed women's right to abortion.
The President announced over the weekend his plan to unveil the Supreme Court nominee the first Monday after the Fourth of July holiday weekend, saying "I will be making my choice for Justice of the United States Supreme Court on the first Monday after the July 4th Holiday, July 9th!"
On Monday, Senate Minority Leader Charles Schumer, D-N.Y. criticized the record of U.S. Appeals Court Judge Amy Coney Barrett of IN, who is one of seven judges considered to be front-runners.
It's true elections have consequences, but the disjunction between the Senate majority that produces a very conservative court, on one hand, and, on the other, the public, derives in large part from lesser-populated red states providing a cushion for Republicans in the Senate (and hence the court) and the electoral college.
One candidate would fail to honor Trump's pro-life pledge, and another would violate Trump's Second Amendment pledge.
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.
"These are very talented people, brilliant people", Trump said Tuesday during an appearance in West Virginia.
Former House speaker and Trump ally Newt Gingrich tweeted Monday: "Judge Amy Coney Barrett would make an outstanding Supreme Court Justice".
Walker said that if it comes down to the vice president casting a tie-breaking vote, he thinks it could tear the country apart.
And while Democrats fear another conservative justice in general, one large area of focus will be on any nominee's stance on abortion. CNN noted that McGahn's reassignment signals he is safe from ousting and will likely remain at the White House, while Shah may resign shortly after the process is complete.
Indiana Sen. Joe Donnelly recently met with Trump to talk about Supreme Court nominees. Eighty percent of Democrats opposed Trump nominating a judge who would make abortion illegal and 66 percent of independents said the same. He has a litmus test for his Supreme Court nominee: end Roe v. Wade.
Several of the women anxious about Trump's tweets and about new restrictions on clinics that perform abortions.