Washington (CNN) The longest US government shutdown ever still has no end in sight.

President Donald Trump's latest proposal -- $5.7 billion in funding for a border wall in exchange for extending temporary protections to immigrants his administration sought to end -- was swiftly rejected by Democrats over the weekend.

But Senate Majority Leader Mitch McConnell said he intended to bring it to the floor this week, a vote that could come Thursday.

"Everyone has made their point—now it's time to make a law," McConnell said Saturday.

But by then House Speaker Nancy Pelosi had already announced her opposition, calling the President's latest proposal a "non-starter."

The public isn't happy with anyone at the moment. A majority of Americans disapprove of the way Trump (61%), Republicans in Congress (60%) and Democrats in Congress (53%) are handling negotiations over the shutdown, according to Pew Research.

What to expect in the Senate

The vote in the Senate on the President's proposal likely won't get the required 60 votes and Democratic support to move forward.

The bill will include the $5.7 billion for the border wall, $800 million for humanitarian assistance and provide three years of legal status for those enrolled in the Deferred Action for Childhood Arrivals and Temporary Protected Status programs.

McConnell will put additional pressure on Democrats to vote for it, introducing not just the President's plan but a full legislative package that will make it harder for Democrats to oppose, including all seven unpassed full-year appropriations measures negotiated by Republicans and Democrats in both chambers, $12.7 billion in disaster relief funding and an extension of the Violence Against Women Act through the end of September, according to a source with direct knowledge of the bill.

Yet the vast majority of Democrats will oppose it and continue to push for the government to be reopened without funding for the border wall.

A Senate Democratic aide, who asked not to be identified, predicted it would not get the 60 votes it needs to advance.

"POTUS proposal can't get 60 votes in Senate and was never designed to," the aide said. "But the White House knew that already."

Democrats will continue highlighting the shutdown's negative effects on the hundreds of thousands of government workers who aren't getting paid. On Tuesday, Sen. Tim Kaine, D-Virginia, will volunteer at chef José Andrés' World Central Kitchen to serve food to some of them.

"If the Pesident wants to negotiate over immigration reform, I'm all for it," said Sen. Elizabeth Warren, D-Massachusetts, on Monday. "I have supported compromised immigration proposals in the past and I'm sure many others will as well. But open the government. Open it now."

What to expect in the House

The House will vote this week on a package of six spending bills that would reopen most federal agencies impacted by the shutdown and include roughly $1 billion in border security-related funding, but not for Trump's demand of a border wall.

The legislation includes $524 million for construction and improvements to ports of entry along the US-Mexico border and $563 million in funding for judges to deal with the backlog of immigration cases.

It has been crafted to reflect agreements previously reached by congressional negotiators on six of the seven remaining appropriations bills for fiscal year 2019 with the exception of the Department of Homeland Security, which has proven to be the most contentious funding bill as a result of the impasse between Democrats and the President over border wall funding.

The effort will be the latest in a series of legislation that Democrats have passed since taking over the House majority in January in an effort to put pressure on Republicans to reopen the government without border wall money.

Those bills have been dead on arrival in the Senate and have faced White House veto threats as a result.

The House has previously passed stopgap funding measures to reopen DHS without allocating any new wall funding since Democrats took over the majority. And House Democrats are likely to once again separately take up a funding measure for the department this week.

The roughly $1 billion in border-related funding in the legislative package will also give Democrats an opportunity to point to examples of the types of border security measures they are willing to support -- as the President continues to press for border wall funding and criticize Democrats for refusing to meet his demand.

The House will also hold another vote on Wednesday on a stopgap funding measure to reopen shuttered parts of the government through February 28. It will give House Republicans the opportunity to undo what they did last week, when they accidentally voted for the proposal.