"We have done it", Pelosi said.
Although the White House initially signaled support for background checks after mass shootings in August, it eventually backed away from that position.
"Mitch McConnell and Republicans in the Senate, you know, they want to abdicate their responsibility here and just point to the president", Van Hollen alleged later in the interview.
Behind the scenes, Senate Democrats and Republicans have engaged in a series of talks with senior White House staff about a package of gun measures that could form the basis of legislation.
Sen. Pat Toomey, R-Pa., who has championed another bill that would expand background checks, told reporters, "It's time to act now".
Trump said conversations would continue Wednesday night and into Thursday, and the senators said they hoped to hear back from the White House then.
Georgia Republican Sen. Johnny Isakson, a conservative who is often considered a voice of the caucus, emerged from Tuesday's lunch saying, "Many of us feel like doing nothing is not a satisfactory answer".
McConnell met Tuesday with a bipartisan group of USA mayors, including some from cities where mass shootings occurred.
Asked if he supported background checks on all private gun sales, the president was non-committal.
The U.S. Conference of Mayors is focusing on background checks as a first step to stem gun violence.
Schumer urged lawmakers from the two parties to vote to repeal the president's plan, saying, "Who else will he take money from and use it for purposes that he wants but that Congress doesn't and the American people are largely opposed to?"
Late Tuesday, the House Judiciary Committee approved several gun bills, including a red-flag law and a ban on large-capacity magazines. "That's quite unfortunate. The NRA clearly has indicated nothing is acceptable to them as it relates to reasonable gun safety measures, even though the Supreme Court in the Heller decision and in an opinion authored by Justice Scalia, of all people, has made clear that pursuant to the Constitution, reasonable gun safety measures are lawful".
The House, which is controlled by Democrats, passed legislation in February that would close loopholes allowing individuals to purchase guns without a background check. McConnell has thus far blocked that bill from facing a vote before the Senate, which is controlled by Republicans, along with other gun reform legislation.
However, contrary to what Schumer has said, polls showed opposition to the border wall is shrinking with 54% at the beginning of this year compared to 63% of the last year, according to the Washington Examiner. More than 10,000 people have died due to gun violence in the country this year, while over 20,000 have been injured.