Schumer said Democrats would "very seriously consider" the offer of a short-term funding deal, while knocking down Trump's latest offer.
Of course, it's not exactly clear what those "other funding sources" would be.
Democrats have refused to sign off on more than US$1.3 billion (NZ$1.89 billion) for security at the border with Mexico. "So that won't happen".
Schumer called the wall "wasteful" and "unnecessary".
One option circulating on Capitol Hill would be to approve government funding at existing levels and kick the issue into the next Congress.
The fact that negotiations are happening at all is a sign that Trump is folding.
Neither he nor his aides would specify whether border security investments must include wall funds.
Shutdown looms as Trump weighs next move on border wall
Trump wants the next funding package to include at least $5 billion for it, an idea Democrats have flatly rejected. Without a resolution, parts of the federal government will close at midnight Friday.
Trump is seen inspecting border wall sample sections in March.
Donald Trump has been forced into an embarrassing retreat after threatening a government shutdown if he could not get funding for his border wall. "We want to shut down the border from illegal immigration".
The chances that a shutdown would be averted went up substantially on Tuesday, when the White House indicated it might accept budget bills without $5bn for a border wall.
Pelosi and Schumer have proposed $1.3 billion for border fencing and other security measures.
Republicans hold a solid majority in the House of Representatives, but only a slim 51-49 majority in the Senate, so Democrat support is needed to meet the 60-vote threshold for overcoming a filibuster.
"I still believe in our Constitution, and Congress is the legislative branch", said Sen.
US President Donald Trump listens to remarks at a roundtable discussion at the White House in Washington, US Tuesday.
As Politico notes, Sanders was likely referring to a compromise bill pushed for by Senate Democrats but rejected by Republicans who wanted the full $5 billion.
James Comey says House Republicans are 'shameful' after interview
On Sunday, he called his former personal lawyer, Michael Cohen , a "rat" because he has cooperated with prosecutors. But I think what he said to Chris Wallace yesterday was not aimed at the public, it was aimed to Bob Mueller.
House and Senate Republicans have been in talks with the White House in recent days looking at other ways to try to secure funding, outside of the traditional appropriations process.
"It's not effective. It's the wrong thing to do and it's a waste of money".
Democratic leaders in Congress, however, refuse to give Trump the money for the wall. But that appeared at odds with Senate Republicans who on Monday told CNN that they felt in the dark about the White House's plans to fund the government.
Meanwhile, the stock market has fallen precipitously in recent weeks, creating economic angst over Trump's agenda.
One option for lawmakers would be to provide stopgap funding for a few weeks, until the new Congress convenes January 3, when Pelosi is poised to become House speaker.
The people spoke on the condition of anonymity to disclose the internal White House sentiment.
President Trump on Tuesday took to Twitter in defense of his desired wall along the U.S. -Mexico border, describing it as an aesthetically-pleasing structure that will also offer the necessary protection for the country. But those efforts never materialized, and he was under heavy pressure to avoid a partial government shutdown just a few days before Christmas. If the case is made to the American people that Republicans should be in control of the House again and Trump is reelected with a GOP majority in 2020, how will anything be different from the past two years of GOP control with no wall funding?
First transgender Miss Universe contestant doesn't win, still makes history
Miss Spain's Angela Ponce became the competition's first transgender contestant, a major step for the 66-year-old pageant. She said that this year's Miss Universe would be a victory for human rights.