WASHINGTON — President Trump attacked Speaker Nancy Pelosi on Sunday for rejecting his proposal to end the partial government shutdown, as he and Republican leaders in Congress sought to put Democrats on defense, a place they have rarely been during the shutdown stalemate.In a series of morning tweets, Mr. Trump said Ms. Pelosi had behaved “so irrationally” in spurning his offer to restore the temporary protections he took away from some undocumented immigrants in exchange for $5.7 billion for a border wall. He also pushed back against conservative critics who called the plan amnesty for immigrants who came to the United States illegally.Senator Mitch McConnell of Kentucky, the majority leader, plans to bring up legislation as early as Tuesday that would wrap Mr. Trump’s proposal into a broader package that would include billions of dollars in disaster relief and immediately reopen the government, an aide said. The move is intended to ratchet up pressure on Democrats, who have insisted they will not negotiate with Mr. Trump until the shutdown is over.“President Trump has put forward a serious and reasonable offer to reform parts of our broken immigration system and reopen government,” Representative Kevin McCarthy, the Republican leader, said on Twitter. “The moment now turns to Nancy Pelosi and Chuck Schumer. The country is watching.”But there was little indication from Democrats, who have denounced Mr. Trump’s plan as “hostage taking,” that they would abandon that position.“If he opens up the government, we’ll discuss whatever he offers, but hostage taking should not work,” Senator Chuck Schumer, the Democratic leader, told reporters in New York on Sunday, adding, “It’s very hard to negotiate when a gun is held to your head.”With the shutdown now in its fifth week, and 800,000 federal employees still furloughed or working without pay, pressure is rising on both Republicans and Democrats to end it. If it lasts until Friday, Congress will be forced to reckon with another grim milestone: Affected workers will miss their second paycheck.[Read: Trump’s Deal Meets With Skepticism Among Immigrants in Texas]As Republicans go on offense in the Senate, Democrats will stay on offense in the House, where they have already passed a string of government funding bills that Mr. McConnell — who has largely remained on the sidelines during the shutdown debate — has refused to take up. On Wednesday, Ms. Pelosi plans to bring up a package of six bills to fund shuttered government agencies, ignoring Mr. Trump’s requests.The measures gained approval last year from Republicans on the House and Senate Appropriations Committees; in effect, Ms. Pelosi will be daring Republicans to vote against their own legislation. Democrats will also begin talking more about their own vision for border security — a shift in strategy for a party that has focused mostly on the economic and personal toll of the shutdown.Amid the posturing, however, there were some hints that Mr. Trump’s proposal, which he unveiled in a 13-minute White House address on Saturday, might open the door to talks that could result in a deal. Vice President Mike Pence, in an appearance on “Fox News Sunday,” suggested that Mr. Trump’s proposal might be amended on the Senate floor.“Well, of course,” Mr. Pence said, when the host, Chris Wallace, asked if he was open to negotiation. “The legislative process is a negotiation.”And Senator Mark Warner of Virginia, a centrist Democrat, described Mr. Trump’s plan as “a starting point” on NBC’s “Meet the Press,” though he added, “You’ve got to start by reopening the government.”Polls show the public largely blames Mr. Trump for the shutdown, and his advisers have been searching for an exit strategy. Last week, Mr. McConnell told Mr. Trump, Mr. Pence and White House advisers that he needed the president to put forth a public offer so he could bring legislation to the floor.“McConnell’s view of shutdowns is when you’re in one, it’s a great opportunity for both sides to get things,” said Scott Jennings, a Republican strategist and former McConnell aide who remains close to him. “And what I think is dumbfounding everyone here is: Why aren’t the Democrats treating it as a situation where they could get something? The president has opened the door for them to get something, so let’s talk.”Throughout his administration, Mr. Trump has sought to limit both legal and illegal immigration. Last year, he moved to rescind Deferred Action for Childhood Arrivals, or DACA, which shielded from deportation hundreds of thousands of young unauthorized immigrants known as Dreamers. He also revoked Temporary Protected Status, or T.P.S., for some undocumented immigrants from Latin American and African countries.In the plan he unveiled on Saturday, Mr. Trump offered to restore the DACA and T.P.S. protections for three years. But Democrats say that is a nonstarter, because it does not offer a pathway to citizenship for Dreamers, and because Mr. Trump is simply giving back protections that he himself took away.The McConnell aide, Don Stewart, said the Republican legislative package would include seven appropriations bills that would fund government agencies that have been partially closed for a month.“The legislation that the majority leader will bring to the floor this week would both reopen the remaining portions of the government, fund disaster relief, fund border security and address immigration issues that both Republicans and Democrats would like to address — all in one bill,” Mr. Stewart said.Mr. McConnell would need votes from all Republicans and seven Democrats for the package to pass. But only a handful of Democrats in Republican-leaning states might feel pressured enough to cross the aisle. A spokesman for one of them, Senator Joe Manchin III of West Virginia, said Mr. Manchin — who just won re-election — would remain undecided on the Republican proposal until he saw it.In broadening the package beyond Mr. Trump’s plan, Mr. McConnell may be trying to pick up additional Democrats — especially those who are eager to vote for disaster relief.“He’s starting to move the pieces around to see what’s going to fly and what’s not going to make it,” said Jim Manley, who was a top aide to Harry Reid, the former Democratic leader from Nevada. “He’s trying to put the pressure back on Democrats, and he’s trying to see what the possibilities of a deal are by putting a larger package there. He’s trying to see how large the package has to be to bring Democrats on board.”But Mr. Trump seemed to be holding fast. He retweeted a series of endorsements of his proposal from Republican lawmakers on Sunday evening, and earlier in the day, he took aim at Ms. Pelosi and pushed back against conservative critics who have described the plan as amnesty for undocumented immigrants.“Nancy Pelosi and some of the Democrats turned down my offer yesterday before I even got up to speak,” Mr. Trump wrote. “They don’t see crime & drugs, they only see 2020 — which they are not going to win. Best economy! They should do the right thing for the Country & allow people to go back to work.”Mr. Trump went on: “Nancy Pelosi has behaved so irrationally & has gone so far to the left that she has now officially become a Radical Democrat. She is so petrified of the ‘lefties’ in her party that she has lost control…And by the way, clean up the streets in San Francisco, they are disgusting!”And to his conservative critics, Mr. Trump wrote: “No, Amnesty is not a part of my offer. It is a 3 year extension of DACA. Amnesty will be used only on a much bigger deal, whether on immigration or something else. Likewise there will be no big push to remove the 11,000,000 plus people who are here illegally — but be careful Nancy!”Ms. Pelosi took to Twitter to strike back. “@realdonaldtrump, 800,000 Americans are going without pay. Re-open the government, let workers get their paychecks and then we can discuss how we can come together to protect the border,” she wrote, using the hashtag #EndTheShutdown.