The Democrats' two-part package includes a bill to fund the Department of Homeland Security at current levels through February 8 and provide $1.3 billion for border fencing and $300 million for other border security items including technology and cameras.
Sen. Lindsey Graham (R-SC), however, warned that Republicans are probably going to have to give some ground on immigration if they are going to get their wall.
In one of the tweets on Monday, Trump reminded Democrats they have voted in favor of a border barrier twice before. But a previous attempt to reach a compromise that addressed the status of "Dreamers" - young immigrants brought to the USA illegally as children - broke down a year ago as a result of escalating White House demands.
They said their goal was to separate Mr. Trump's demand for a border wall from the ongoing government shutdown, reopening government while putting off a fight over homeland security's budget for several weeks.
Talks between the White House and Democrats broke down last week and now House Democrats are planning to use their new majority on Thursday to vote on legislation to end the shutdown without adding funds for the wall.
And it does not include the $5 billion Trump has demanded, instead allocating $1.3 billion for border security and fencing.
The shutdown began December 22 and there has been no sign of direct negotiations involving Republicans, Democrats and the White House. After the Senate easily passed the measure without any money for the wall, Trump rejected it in the wake of criticism from conservative pundits, voters and lawmakers. "We have to have border security and the wall is a big part of border security".
The ducks are lined up something like this: Donald Trump seems to be enjoying the shutdown, he is setting the agenda, politicos are talking about what he wants them to talk about and because they are doing this, Trump can keep tweeting about what he wants to tweet on the border wall, fueling his base and distracting from the looming Mueller probe finale that may hit any day now. "Sovereignty is the defining issue of President Trump and his administration", Malkin said on "Fox & Friends" Tuesday.
"Responsibly funding the federal government is one of the most important duties of Congress", House Appropriations Committee Ranking Member Nita Lowey (D-N.Y.), who introduced the bills, said in a statement provided to LifeZette.
Pelosi, who is expected to be elected speaker this week, has called the wall "immoral" and said her new majority will not pay for it. And they have said they need barriers.
The White House has not commented on the lawsuit.
The government shutdown is now stretching into 2019.
Trump on Twitter criticized Democratic opposition to the wall project, which carries a total estimated price tag of $23 billion.
Graham had also gone to the White House to complain about Trump's abrupt decision to declare victory over ISIS and remove US troops from Syria. They hold the purse-strings of the USA federal government now.
"Once the Senate passes this legislation and puts us on a path to reopening government, the president must come to his senses and immediately sign it into law". Is he willing to accept anything that does not include money specifically to build a new border wall, even if the Democrats, as they have said, are willing to get money for more border security?