By SHERYL GAY STOLBERG and EMILY COCHRANE, New York Times–

Senator Mitch McConnell, the majority leader, conceded on Monday that he could not stave off final passage of a resolution overturning President Trump’s national emergency declaration, setting up a rebuke to Mr. Trump amid signs that the president’s grip even on his own party in Congress may be slipping.

With Senator Rand Paul of Kentucky joining three other Republicans — Susan Collins of Maine, Lisa Murkowski of Alaska and Thom Tillis of North Carolina — in announcing he would support the measure, Democrats now have the 51 votes they need to secure passage and to force Mr. Trump to issue the first veto of his presidency.

Mr. McConnell is exploring whether he can amend the House-passed resolution of disapproval, to send it back to the House and slow its trip to the president’s desk. Ms. Collins and Ms. Murkowski are both sponsors of a separate resolution, virtually identical to the House resolution, introduced in the Senate last week.

And while a veto is highly unlikely to be overturned, the congressional majority that forces it will stand as a powerful rejection of the tactics Mr. Trump has used to fulfill his top campaign promise to build a wall on the southern border — and will apparently be the first time since passage of the National Emergencies Act of 1976 that Congress has voted to overturn an emergency declaration.

“It simply sends a message that Congress is going to stand up for its institutional prerogatives and abide by the separation-of-powers framework that was carefully worked out by the framers in the Constitution,” Ms. Collins said on Monday. “I truly don’t see this as sending a message at all one way or the other about border security but rather about executive overreach.”

Ross K. Baker, a political scientist and expert on Congress at Rutgers University, said passage of the resolution would amount to a “serious rebuke” of the president.

“It’s Congress saying: ‘This has gone far enough. We’re not going to roll over and play dead for the president,’” Mr. Baker said. “It’s the kind of thing that James Madison had in mind when he laid out in the Federalist Papers the argument for separation of powers and checks and balances.”

Mr. McConnell has not set a date for the vote, but it is expected before March 15, when lawmakers leave for recess; Mr. Paul said on Monday that he believed as many as 10 Republicans would vote for the resolution. Once Mr. Trump issues his expected veto, the matter is all but certain to be settled in the courts, where multiple lawsuits have been filed.

Speaking at an event in his home state of Kentucky on Monday, Mr. McConnell offered a matter-of-fact assessment of the reality facing the White House: “I think what is clear in the Senate is there will be enough votes to pass the resolution of disapproval, which will then be vetoed by the president and then in all likelihood the veto will be upheld in the House.”

Some Republicans are trying to find another way out. Senator Lamar Alexander, Republican of Tennessee, has urged the president to reconsider using military construction money to fund the wall.

He proposed that Mr. Trump take the additional money from the Treasury Forfeiture Fund and the Defense Department accounts used to support counterdrug activities and efforts to block drug smuggling — congressionally approved funds that the president already plans to use. But Mr. Alexander told reporters on Monday that he had not spoken with the White House about his idea.

Other senators, including Mr. Paul, have raised the possibility of amending the National Emergencies Act to curtail the powers available to the president and give Congress more oversight over a declaration.

Congress has long bristled at what the historian Arthur Schlesinger Jr. branded the “imperial presidency” — a reference to Richard M. Nixon’s actions during Watergate and the Vietnam War.

Democrats fumed at what they viewed as President George W. Bush’s expansive use of his executive powers; Republicans routinely accused President Barack Obama of exceeding his authority on issues like combating climate change and protecting certain classes of undocumented immigrants.

But Mr. Trump’s declaration of a national emergency to fulfill a campaign promise to build the wall — issued after Congress denied him the money for it — strikes many lawmakers as a direct incursion on a power granted exclusively to Congress in the Constitution: the power of the purse.

“Congress has the power of the purse. Congress is a check on the executive,” Senator Chuck Schumer of New York, the Democratic leader, said in a speech on the Senate floor on Monday. “The founding fathers feared probably above all anything else, having dealt with King George in the Revolution, that an overreaching executive was one of the greatest dangers to our democracy.”

Mr. Paul said as much in an opinion piece published late Sunday on Fox News’s website, and again on Monday.

“It’s not like I sat up one day and said, ‘Where do I want to separate myself from President Trump?’ There are political risks to separating yourself from your own party’s president,” he said, adding, “To me, it isn’t even about immigration, it isn’t about a Republican or Democratic president.”

“It’s about Congress versus the president and where the power should be distributed,” he concluded.

In his opinion piece outlining his vote, Mr. Paul went so far as to speculate that Justices Neil M. Gorsuch and Brett M. Kavanaugh — both conservative judges appointed by Mr. Trump to the Supreme Court — would join the majority in ruling against the administration and the national emergency declaration.

“Without question, the president’s order for more wall money contradicts the will of Congress and will, in all likelihood, be struck down by the Supreme Court,” Mr. Paul wrote. “In fact, I think the president’s own picks to the Supreme Court may rebuke him on this.”

Mr. Trump’s emergency declaration — the first time a president has invoked powers under the National Emergencies Act after Congress denied funds — is a particularly thorny issue for Mr. McConnell.

Like many of his conservative colleagues, he warned the president against setting a precedent that future Democratic presidents could seize as a means for carrying out stringent gun control policies or climate change controls. But it was Mr. McConnell, in a speech on the Senate floor, who announced Mr. Trump’s intent to declare a national emergency — and his support for the president’s decision.

On Monday morning, the White House dispatched Zach Parkinson, the deputy director of government communications, to meet with Republican communications aides, according to two people in the meeting, who asked for anonymity to discuss a private meeting. Mr. Parkinson asked the aides to “keep their powder dry” ahead of the vote, and urged them to have their senators reach out to the White House Office of Legislative Affairs with concerns.

Less than half an hour later, Mr. McConnell acknowledged at a news conference in Kentucky that he had “argued, obviously without success, that the president not take this route” because of concerns about setting a precedent future Democratic presidents could take advantage of.

Mr. McConnell said he was involved in discussions with the Senate parliamentarian to see if the one-page resolution that passed through the House could be amended. Both Ms. Collins and Ms. Murkowski have emphasized that their endorsement is for a measure that does not contain additional amendments or more pointed language.

“It’s an interesting question,” Mr. McConnell said in Kentucky, referring to whether he could amend the House language. “It’s never been done before.”