But Democrats largely oppose Trump's proposed wall, which he had initially said would financed by Mexico. They have offered support for $1.3 billion in general border security funding. It was not clear if some compromise could be struck between that offer and Trump's demand.
Over the weekend, Mick Mulvaney, the acting White House chief of staff, said the White House had made a counter-offer to Democrats on border security. Media reports said Vice President Mike Pence had proposed $2.1 billion in funding.
Last week Trump said his administration was prepared for a lengthy shutdown.
After weeks of failed talks between Trump and congressional leaders, parts of the U.S. government shut down on Saturday, affecting about 800,000 employees of the Departments of Homeland Security, Justice, Agriculture, Commerce and other agencies.
Most of the federal government, which directly employs almost 4 million people, is unaffected. The Defense, Energy, Labor and other departments are funded through Sept. 30.
Even agencies that are affected never totally close, with workers deemed "essential" still performing their duties.
"Non-essential" federal workers at unfunded agencies are on furlough and staying home. Both they and essential employees will not get paychecks after December until the shutdown ends.
"We continue to believe that it is unlikely that Congress will come up with a deal to end the current partial shutdown until well into January," financial firm Height Securities said in a commentary note on Wednesday.
The 435-seat House was set to reopen on Thursday but on Jan. 3, the 2017-18 Congress will be replaced by the 2019-20 Congress and control of the House will switch to the Democrats from the Republicans. At that time, Representative Nancy Pelosi is expected to take over as House speaker.
She has vowed swift action to fully reopen the government. Barring some sort of deal in the interim, House Democrats expect to vote on a funding bill on Jan. 3, a Democratic aide said.
In the new Congress, Senate Republicans will increase their number of seats to 53 but still will need Democratic support to pass any legislation requiring a 60-vote majority.
Details of the upcoming House bill were unclear but it was unlikely to include wall funding, like an earlier Senate measure. If such a bill were to pass the House and again win support in the Senate, it would then go to Trump.
At that point, he could face a politically difficult choice - back down on his full wall-funding demand or veto the bill and single-handedly extend the partial shutdown.
If he chose the latter, putting his personal stamp on the shutdown, Congress might then move to override his veto, but that would take a two-thirds vote in both the Senate and the House, a challenging hurdle for lawmakers.
(Reporting by Kevin Drawbaugh and David Morgan; Editing by Richard Chang and Bill Trott) ((email@example.com; +1-202-898-8314; Reuters Messaging: firstname.lastname@example.org))