The president's acting chief of staff, Mick Mulvaney, said that the offer was a concession to Democrats in Congress who have refused to give Mr Trump $5.6 billion to begin building a $23 billion barrier on the border with Mexico.
"This is a very important battle to win, in terms of safety, number one, defining our country, and who we are", Trump said.
Democrats have said they will only discuss border security once the government is reopened. "She went in front of an enthusiastic crowd of her supporters and it was red meat for them. Good solution, and made in the U.S.A".
About 800,000 federal workers have been furloughed or are working without pay.
She explained that the goal of Friday's White House meeting was to get the government reopened.
In his interview on NBC, Mulvaney conceded that Trump, who said last month he would "take the mantle of shutting down" the government over border security, is losing the fight over public opinion over the shutdown and accused Democrats of trying to stall.
White House officials and congressional aides will meet again Sunday to continue discussions about how to end the partial government shutdown that has affected hundreds of thousands of federal workers and caused delays and inconvenience for Americans.
Senior White House adviser, and Trump son-in-law, Jared Kushner, and Homeland Security Secretary Kirstjen Nielsen were involved in the meeting Saturday, with Nielsen briefing those in attendance on border security.
"I may decide a national emergency depending on what happens over the next few days", Trump told reporters at the White House before leaving for Camp David, a presidential retreat in the state of Maryland, for meetings with senior White House staff on border security and other issues.
"It has to be a bipartisan process", Schiff said on CNN's State of the Union.
Trump has demanded Congress appropriate money for the wall, and his dispute with Democrats over the issue pushed the government into an ongoing partial shutdown.
Trump said Friday he could declare a national emergency to circumvent Congress and build the wall, though budget experts said Congress would still need to allocate the funds.
President Donald Trump says he plans to call the heads of U.S. Steel and other steel companies to come up with a new design for the barrier he's insisting must be built along the southern border.
Republican Susan Collins of ME wants the Senate to take up House-passed bills to re-open the government.
"Are you recalibrating your assessment of how you can work with this president?" Even if the House advances any articles of impeachment, a two-thirds-majority vote to convict Trump in the Republican-controlled Senate and remove him from office would seem out of the question, barring astonishing new revelations.
Mr Trump is demanding that any funding to keep the government operational also includes money to help build the wall.
"The House will first consider the Financial Services and General Government appropriations bill, which funds the Department of the Treasury and the Internal Revenue Service, to ensure that hardworking Americans can receive their tax refunds on schedule".
"Republican hypocrisy at its finest: saying that Trump admitting to sexual assault on tape is just 'locker room talk, ' but scandalizing themselves into faux-outrage when my sis says a curse word in a bar", Ocasio-Cortez wrote Saturday on Twitter, referring to Trump's lewd comments on a 2005 "Access Hollywood " tape that surfaced during the 2016 campaign.