It also includes $1.37 billion in new money to help build 55 miles (88.5 km) of new physical border barriers.
Under the tentative agreement reached by Congress this week, funding for enhanced security on the border would be limited to $1.375 billion.
Mr Trump sparked the last shutdown before Christmas after Democrats snubbed his $5.7-billion demand for the wall.
The funds would cover just part of the estimated $23 billion cost of the wall promised by Trump along the 2,000-mile (3,200-km) border with Mexico. They point out that Democratic House speaker Nancy Pelosi has the power to force the Senate to vote on a resolution to rescind the declaration, which if it does not succeed could extract a potentially large political cost from Republicans.
The bill, passed overwhelmingly by both the U.S. Senate and House of Representatives on Thursday, contains money for fencing and other forms of border security. The bill does not provide money for President Donald Trump's border wall, but does contain money for other border security measures.
Mitch McConnell said he would be supporting the President's move. "The President is once again delivering on his promise to build the wall, protect the border, and secure our great country".
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Pelosi said this move by Trump is just him overstepping his bounds to navigate around Congress.
Velázquez, who was born and raised in Puerto Rico, is concerned that Trump may use the national emergency declaration to take money from the relief effort to help the US territory with its recovery from the 2017 hurricanes that ravaged the island. "President Trump, sign this bill", Schumer said.
Julian Castro, the former San Antonio mayor who was President Barack Obama's Housing and Urban Development (HUD) Secretary, said Trump's intention to declare a national emergency at the border is "appalling" and would "amount to an abuse of power that should be immediately challenged in the courts".
"The reason for that is as with so much else Trump, what's he's doing would really smash through another constitutional guard rail in the sense that he would be explicitly doing an end run around Congress". Democrats also see victories in increased funding for humanitarian aid at the border, alternatives to detention and aid to Central America.
The legislation would also fund the Justice Department, Commerce Department, State Department, Department of Agriculture, Internal Revenue Service and others, covering roughly 800,000 federal workers.
Simply having the money available would not resolve potential roadblocks to building the wall on the local level, the aides said.
Other Republicans said they were hoping for assurances Mr Trump will sign the bill if it gets to his desk, possibly later on on Thursday.
About $10 billion would be available to the president from this year's budget, along with another $11 billion in funding from previous budgets that has not been obligated, the aides said.