Conway claims "the president has already compromised" by dropping his request for the wall from $25 billion to $5 billion. He also said the compromise should include changes in federal law to discourage people from trying to enter the USA illegally.
There has been little direct contact between the sides during the stalemate, and Trump did not ask Republicans, who hold a monopoly on power in Washington until January 3, to keep Congress in session.
"Of course we will always defend our sovereignty ..."
Huckabee advised Trump to do more than tweet, and instead appear in a nationally televised address to "lay out the whole situation".
Trump visited San Diego in March to review border wall prototypes commissioned by the Department of Homeland Security, but he has not visited a Border Patrol station or USA port of entry.
Trump, who campaigned on building a "big, handsome wall" that Mexico will pay for, has recently backed off his desire for a concrete barrier, instead signaling he would be willing to opt for "steel slats".
The White House said the Democrats were provided with a proposal, but haven't responded.
As more agencies run out of funding, President Trump is now also threatening to close the border with Mexico if Democrats refuse his demand for money to build a wall along the southern border.
Days later United States authorities dramatically shuttered a border crossing in southern California after hundreds of migrants - part of the "caravan" Trump had roundly condemned - tried to breach a fence from the Mexican city of Tijuana.
No progress is being reported on ending the partial closure that began December 22 and entered its seventh day Friday.
The agreement, agreed to by Canada, the USA and Mexico under president Bill Clinton in 1994, enabled the free-flow of goods between all three countries.
The president met with his chief shutdown negotiators Friday night.
Trump canceled his Christmas trip to Florida and remained in the White House, saying on Christmas Eve that he was "waiting for the Democrats to come back and make a deal".
"This is not Pelosi's first holiday vacation to Hawaii".
The President has repeatedly voiced his disdain toward migrants from these countries who have sought to cross into the USA from Mexico outside of official ports of entry.
Schumer is the top Democrat in the Senate.
Mulvaney argued that if Pelosi reached an agreement with the president, she would likely loose her leadership of the House of Representatives.
Rejecting next week's spending measure from House Democrats would put a Republican stamp on the shutdown.
Polls say otherwise, Trump himself is walking back his demands.
Pelosi and Schumer have been discussing three options for Democratic legislation, according to a senior Democratic aide.
- Pass the stopgap spending bill that the Senate passed, which would fund all government agencies through 8 February at their current levels.
At the weekend, Trump was tweeted often, but did not hold meetings with Republican or Democratic leaders.
The shutdown affects about 800,000 employees.
Most of the federal government, which directly employs nearly four million people, is unaffected.