"Partisan games and shutdown threats are no way to run our government, and I hope House Republicans will be willing to stand up to President Trump and work in good faith to produce appropriations bills with bipartisan support", said Nita Lowey of NY, the top Democrat on the House Appropriations Committee, in a statement.
Beyond building a wall, the administration also wants to do away with so-called "chain migration" allowing relatives of immigrants to come to the country, and the diversity visa-lottery program, and to move toward what Trump calls a "merit-based" immigration system. McConnell acknowledged, however, that the border funding issue in particular was unlikely to be resolved before the balloting.
Trump campaigned on the promise of building a wall to deter illegal immigration and making Mexico pay for it. Mexico has refused, leading Trump to look to US taxpayers to fund the endeavor instead, at least for now.
Trump has asked for $25 billion to build the wall.
While some Republicans offered measured criticism of the problem, Democrats ridiculed Trump's immigration policies as cruel and bungling. The House is now in recess, returning after Labor Day.
Congress ultimately passed a spending bill in March that funded the government through September.
Trump would be taking a political risk if he allows most government functions to lapse on October 1 - the first day of the new budget year - roughly a month before the elections, when Republican control of both the House and Senate is at stake.
As Greek Wildfire Closed In, a Desperate Dash Ended in Death
Forest fires are common in Greece during the hot, dry summers, and temperatures recently hit highs up to 40 degrees Celsius. A fire brigade spokeswoman said at least 74 people were killed in the wildfires and the death toll was expected to rise .
Both Speaker of the House Paul Ryan and McConnell met with Trump last week to discuss funding the government.
Trump has previously floated the possibility of a government shutdown over border security and immigration, and on Sunday he made his threat explicit, saying he would do so unless Congress funds his proposed wall, which he promised Mexico would pay for, and puts in place his preferred immigration policies. The president tweeted over the weekend that he would be willing to shut down the federal government if Democrats don't support a plan to boost border security and fund his signature border wall.
Earlier, Donald had threatened to shut down the government after his priorities in congressional spending bills was sidelined.
Democrats have long opposed financing Trump's wall but lack the votes by themselves to block House approval of that amount. They have the strength to derail legislation in the closely divided Senate. The House Appropriations Committee Homeland Security Bill contains $5 billion for "physical barriers and associated technology on the southern border".
"I would have no problem doing a shutdown", Mr Trump said during a joint press conference at the White House with Italian prime minister Giuseppe Conte.
"We raised a number of concerns in the ORR programme about any policy which would result in family separation due to concerns we had about the best interest of the child", White told a Senate Judiciary Committee hearing that is looking into how the policy evolved. Richard Durbin, Albence says the agency has records documenting decisions by hundreds of migrant parents to leave the USA without their children. No. 2 Senate Democratic leader Richard Durbin of IL said Homeland Security Secretary Kirstjen Nielsen should resign and that someone "has to accept responsibility" for policies that show "the extremes this administration will go to".
President Donald Trump says there are "consequences when people cross our Border illegally" and claims many who do so are "using children for their own sinister purposes".