In one ofTrump's biggest 2016 election campaign promises, the businessman pledged to slash tax rates for businesses and people who pay taxes, saying that doing so would boost the economy and help create millions of new jobs.
Sen. Mario Rubio (R-Fla.) said the message sent by disarray in Washington would be risky at a time of high tensions over North Korea and other worldwide hot spots.
She added, "The wall is, in my view, immoral, expensive, unwise, and when the president says, 'Well, I promised a wall during my campaign.' I don't think he said he was going to pass billions of dollars of cost of the wall on to the taxpayer".
"The Department of Treasury, several years ago, under the Obama administration, said that if you change the regulations and enforced it properly, you would save up to $4 billion a year", he said.
In a pair of tweets Sunday, the president accused Democrats of withholding money from the federal budget to start building the wall he wants constructed along the USA southern border with Mexico.
But so far, negotiations have proven hard, with disputes over the border wall and health law subsidies to help low-income people afford health insurance.
In a subsequent tweet, he wrote: "Eventually, but at a later date so we can get started early, Mexico will be paying, in some form, for the badly needed border wall". Legislation will require support from Democrats to clear the Senate.
Democrats said they vigorously oppose any money for the border wall in a new spending bill, setting the stage for a last-minute showdown as the White House and lawmakers scramble to pass a stopgap bill before funding expires at the end of Friday.
Figures vary on the cost to build the wall.
Democrats accused Mr Trump of making "poison pill" demands and said he should "back off". The government will shut down at midnight Friday if Congress can not agree on one.
Senator Richard Durbin, a Democrat from IL, said: "To think he would consider shutting down the government over this outlandish proposal of a wall, that would be the height of irresponsibility".
Even with contentious negotiations ahead in the coming days, White House budget director Mick Mulvaney said, "I don't think anybody foresees or expects or wants a shutdown at the end of next week". Mulvaney said talks between Republicans and Democrats could produce an agreement as early as Sunday. "We'll negotiate with the Democrats". He added that "it'll be enough in the negotiation to move forward either with construction or the planning.to get going on the border wall and border security".
Trump has repeatedly asserted that Mexico would pay for the wall, which he says is necessary to stop the flow of immigrants crossing the border illegally as well as drug smugglers.
Priebus said the administration expects "the priorities of the president to be reflected" in a budget.
Sessions conceded he does not expect the government of Mexico to "appropriate money", but maintained the United States has other options to get money from Mexicans.