Pressure mounted on Trump as the shutdown was threatening the economy and left hundreds of thousands of federal workers missing a second paycheck on Friday.
The Republican-led US Senate had earlier rejected two shutdown-ending bills on Thursday.
The Democrats in the House had demanded to reopen the government before any negotiations with Trump and his Republican allies in Congress on border security.
However, the situation worsened on Friday, when the Federal Aviation Administration reported flight stoppages and flight delays at international airports in New York, Philadelphia and Newark, among others.
In recent weeks, Trump had dug in on his demand for a wall, but leading Democrats refused to budge, offering funding for border security other than a wall.
Later on Friday, Americans woke up to an unexpected announcement from Mr. President, who said the US does not need a concrete border wall along the entirety of the southern border.
“We do not need 2,000 miles (3,200km) of concrete wall from sea to shining sea. We never did,” Trump said.
“We never proposed that,” he added.
“We never wanted that because we have barriers at the border where natural structures are as good as anything that we could build.”
Trump added, “Our proposed structures will be in predetermined, high-risk locations that have been specifically identified by the Border Patrol to stop illicit flows of people and drugs.”