Democratic senators have discussed in recent weeks that if they cannot secure the 67 votes needed to convict Trump -- and bar him from holding office in a subsequent simple-majority vote -- that they might invoke the 14th Amendment of the Constitution to do the same.
Schumer, D-N.Y., was asked about the possibility in a press conference ahead of the impeachment trial proceedings Thursday.
"We're first going to finish the impeachment trial and then Democrats will get together and discuss where we go next," Schumer replied.
He'd also praised the case made by the House impeachment managers Wednesday and that he is "hopeful it will change minds. It's hard to look at that and not see the gravity of what happened."
Sen. Tim Kaine, D-Va., has been on the forefront of the 14th Amendment push among Democrats, and Sen. Chris Coons, D-Del., late last month called it "intriguing."
"What Senator Kaine is talking about is a censure resolution that would also specifically include the elements of the 14th Amendment that lead to disqualification from future office," Coons said. "That's intriguing to me and something I'm willing to look at the bottom line here is we have to deliver accountability for the events of January 6."
The 14th Amendment to the Constitution says that Congress can bar people who "engaged in insurrection or rebellion" against the U.S. from holding office. It was originally meant to prevent former Confederates from serving in the government after the Civil War.
"No person shall be a Senator or Representative in Congress, or elector of President and Vice President, or hold any office, civil or military, under the United States, or under any state, who, having previously taken an oath, as a member of Congress, or as an officer of the United States, or as a member of any state legislature, or as an executive or judicial officer of any state, to support the Constitution of the United States, shall have engaged in insurrection or rebellion against the same, or given aid or comfort to the enemies thereof," the amendment says.
"The Congress shall have power to enforce, by appropriate legislation, the provisions of this article," it adds.
Some interpret this amendment as fundamentally giving Congress the ability to bar a person who "engaged in insurrection" -- what Trump is accused of in the article of impeachment -- from office.
It's almost certain that this action, like an impeachment conviction, does not have the votes to pass.
It would likely raise constitutional questions and slippery slope concerns that it would set a precedent neither party wants when their preferred candidate is in office.
Trump's current impeachment was spurred by the Jan. 6 riot at the U.S. Capitol. Trump, after months of making false claims that he'd won the presidential election, called a rally in Washington, D.C., with his supporters for the same day Congress and then-Vice President Mike Pence were meeting in a joint session to certify the results of the election.
Trump, at the rally, repeated his false election claims and he and advisers used pitched rhetoric, riling up the large crowd. Trump at one point in the rally told his followers to "peacefully and patriotically" march to the Capitol, a comment his defenders point to as part of the reason why he does not bear responsibility for the ransacking of the Capitol.
But House impeachment managers are arguing that one comment does not cancel out the balance of Trump's other comments in that speech or in the proceeding months.
"In a speech spanning almost 11,000 words -- yes, we did check -- that was the one time, the only time, President Trump used the word peaceful or any suggestion of nonviolence," impeachment manager Rep. Madeleine Dean, D-Pa., said.
Schumer made his comments at a press conference with Sens. Jon Ossoff, D-Ga., and Raphael Warnock, D-Ga. Democrats more generally but Ossoff, Warnock and Schumer specifically have been the subjects of harsh criticism from progressives for allegedly backing off their campaign promise of $2,000 stimulus checks for Americans.
The coronavirus relief plan being pushed by President Biden and congressional Democrats includes $1,400 stimulus checks, which Democrats say adds up to $2,000 with the $600 stimulus checks passed in the waning days of the Trump administration.
The senators argued that the Democrats' plan in all provides average families with much more than $2,000 in total federal aid.
"Senator Warnock and I are here to deliver for Georgia families who are counting on us for aid during this pandemic," said Senator Ossoff. "This bill will send $8,200 in new federal financial support to an average working family of four in Georgia while we invest massively in vaccines and the health response to end this pandemic."
Fox News' Kelly Phares contributed to this report.