At his news conference Wednesday, Trump said "we've met" with numerous candidates, and indicated he's working off a list of 20 potential nominees he released during the campaign.
"I have a list of 20. I have gone through them. We've met with numerous candidates. They're outstanding in every case," Trump said.
Transition officials did not say whether Trump himself has met with potential court nominees.
"I'll be making the decision on who we will put up ... that will be probably within two weeks of the 20th," Trump said.
Later Wednesday, Pence met with Democratic Sens. Joe Manchin of West Virginia and Joe Donnelly of Indiana in part to try and get them on board for Trump's pick.
A Senate Democratic staffer said it was clear Pence is trying to feel out where moderate and vulnerable Democrats are on the nomination fight, as part of a push to get to a filibuster-proof 60 votes.
"Today was really about talking about our legislative agenda, but also meeting with members of the Senate to get their input on the President's decision about filling the vacancy on the Supreme Court," Pence told reporters after a series of meetings with senators.
Pence also met with former Democratic vice presidential nominee Tim Kaine, Judiciary Committee Chairman Chuck Grassley and GOP Sens. Lisa Murkowski and Susan Collins.
Who's the pick?
Pence echoed Trump in saying there has not been a selection yet, but they are "winnowing" down the list of potential nominees.
The list is made up of mostly federal appellate court judges including Judge William Pryor, Diane Sykes, Steven Colloton, Neil Gorsuch, Raymond Kethledge and Thomas Hardiman.
Manchin said he and Pence did not talk about a specific potential nominee, but was told it would be from the list of 20.
"If you're a Democrat, you can assume it's going to be somebody conservative. That's a given," Manchin said.
"There's been some of the people on that list who have already gone gone through the process here as far as approving. I guess they would look at someone who has gone through, somebody who's made it through here before would have a chance."
Such a declaration could cut out Joan Larsen, who sits on the Michigan Supreme Court, rather than the federal bench. It could also mean Trump won't go with the politician previously floated: GOP Sen. Mike Lee of Utah.
The Federalist Society's Leonard Leo, Heritage Foundation President Jim DeMint and some senators are involved in the process, Trump said.
Sources say that the goal is to get a new justice on the bench by the last sitting of the court which begins on April 17. That would likely require hearings sometime in March.
Democrats are gearing up for the hearings and they refer to the seat as being "stolen" by Senate Republicans who declined to hold hearings fro President Obama's nominee Merrick Garland.
"Let's see who they nominate," Senate Minority Leader Chuck Schumer said last week to reporters. "If they're in the mainstream we'll give them a very careful look. If they're out of the mainstream we'll oppose them tooth and nail."