Trump said Friday that he wanted negotiations for border security to continue ahead of the February 15 deadline, and threatened to use the "very powerful weapon" of declaring a national emergency if negotiations came to naught.
Schumer answered McConnell with the proposal that eventually would be accepted: a three-week measure to reopen the government and then a plan for House and Senate negotiators to discuss border security.
In California for a meeting of the Koch political network, Trump supporter and Koch donor Doug Deason of Texas said he was "severely disappointed" that the president agreed to reopen the federal government.
In a tweet on Saturday, Mr Trump said negotiations between Democrats and Republicans would start immediately but would not be easy because both sides were "very dug in".
The fissures among exasperated members of his party - coupled with security concerns with the freeze on resources raised by the Federal Bureau of Investigation and other agencies - led Trump to conclude he had run out of time and had to reopen the government. "I think she deserves it", he said then.
"I don't see this as any power play", Pelosi said. "And that is what maybe the president underestimated".
Analysts say this could divert military funding towards building the border wall but would provoke constitutional uproar and legal challenges.
As negotiations restart, Mr Trump enters them from a weakened position.
On Friday, a new survey by the Washington Post and ABC News showed that just 37 per cent of Americans approved of how Trump's handling his job as president, while some 53 per cent blame him and congressional Republicans for the shutdown.
But the US President warned his battle with the Democrat-led house of representatives is not over and said the government would shut down again if "we don't get a fair deal from Congress".
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Trump told reporters at the White House that the Democrats have become radicalized and what they are trying to do is a disgrace. Secret Service had told him there would be "absolutely no problem" with security for the evening address on Capitol Hill.
The president has said he could declare a national emergency to fund the border wall unilaterally if Congress doesn't provide the money. "Trump has fairly broad power to do so, though it would nearly certainly be challenged in the court system". Trump said a bipartisan congressional conference committee will meet to come up with a plan for border security.
Previously, he had tweeted that he would not cave to the Democrats on the issue of border wall funding.
"[They have] finally and fully acknowledged that having barriers, fencing ... will be an important part of the solution", Trump said. The deal includes back pay, which the administration promises to get out as soon as possible.
The President has also left open the possibility he will declare a National Emergency to build the wall if a funding deal can not be reached.
However, it would not take place on January 29 as once planned, said a person familiar with the planning, who was not authorized to discuss it.
The president was still adamant about funding more physical barriers on the border. Ann Coulter, the conservative commentator, suggested Trump was "the biggest wimp" to hold the office. "The ball is now in Nancy Pelosi and Chuck Schumer's court, " Stein said. Asked about Trump's wall, Pelosi, who has said repeatedly she won't approve money for it, said: "Have I not been clear?"
Mr Trump also said he would not yet resort to "a very powerful alternative" - an apparent reference to declaring a national emergency.
Friday's announcement was an extraordinary comedown that left many in the White House and those who support Trump marveling at the futility of the preceding four weeks of brinkmanship. The president's approval numbers had suffered during the impasse.
In one of the many effects of the shutdown, hundreds of flights were grounded or delayed at airports in the NY area and Philadelphia on Friday as more air traffic controllers called in sick. But several senators said they didn't know what to expect as they arrived to watch the president's televised address from their lunchroom off the Senate floor.
Trump triggered the shutdown, which began on December 22, when he demanded the $5.7 billion in money for a wall along the U.S. The Democrats' bill won more votes than the GOP bill, even though Republicans control the Senate. It also created a humanitarian crisis as some 800,000 federal employees did not receive their paychecks for a month.