With dozens of Dreamers looking on from the viewing gallery, the Senate was unable to muster the votes Thursday to approve an amnesty for them and 1.8 million other illegal immigrants, leaving their fate in doubt with just weeks to go before many of them start to lose their DACA protections.

President Trump hastened the defeat of one bipartisan plan after he issued the first firm veto threat of his tenure, saying the proposal to couple an amnesty with $25 billion in border wall money didn’t do enough to stop future illegal immigration.

Senators also shot down a more generous amnesty plan from Sen. John McCain, Arizona Republican, and a more enforcement-minded Republican proposal that would have coupled a 1.8 million-person amnesty with the stiffest immigration crackdown in more than two decades.

None of the plans came close to clearing the 60-vote threshold senators set.

“President Trump has stood in the way of every proposal that could become law,” said Senate Minority Leader Charles E. Schumer, New York Democrat. He moved quickly to pin blame on a White House that had set parameters for the debate and watched as lawmakers kept flouting them.

Some senators said they may try again, with a March 5 deadline still looming. The House could also take the lead — though Republican leaders there were struggling to build support for a conservative bill that would grant a limited legal status to 700,000 people who were part of the Obama-era Deferred Action for Childhood Arrivals program in exchange for major changes to legal immigration, a crackdown on sanctuary cities and requirements that businesses do more to verify their employees’ work status.

Lawmakers also speculated that they may revisit the issue as part of the next round of spending bills, due by March 23.

“This does not have to be the end of our efforts,” said Senate Majority Leader Mitch McConnell, Kentucky Republican. “I would encourage members to put away the talking points and get serious about finding a solution that can actually become law.”

Mr. McConnell had promised a roaring debate on immigration and tried to make good on the promise by carving out this week for what all sides had said would be serious floor speeches, dozens of amendments and important votes.

Instead, Democrats gummed up the system by preventing votes and hoping for a bipartisan bill to emerge from closed-door negotiations.

That plan emerged late Wednesday with eight Republicans and eight members of the Democratic caucus, and was offered by Mr. Schumer as Democrats’ ante in the debate. The plan coupled the amnesty for 1.8 million people with funding for a border wall, but it did not tackle the Diversity Visa Lottery, nor did it impose substantive limits on the chain of family migration. It did attempt to prevent Dreamers from sponsoring their parents for legal status, but the administration said that was unworkable and likely illegal.

An even bigger problem, the Homeland Security Department said, was language in the plan that would have reset enforcement priorities, effectively ordering authorities to ignore any illegal immigrants who are already in the U.S. or who can make it into the country by June 30.

That “future amnesty” would protect perhaps 10 million illegal immigrants from deportation, a senior administration official said. He also predicted smugglers would soon be using the plan as an advertisement to spur another surge of illegal immigration.

“We have not seen before a bill that places a future amnesty date in law — a date that is basically a beacon for millions of people around the world who would seek to come here illegally,” the official said.

The White House issued a firm veto statement, which the legislation’s backers said likely chased away some Republicans

“We thought we were within two or three, or maybe had the requisite number, but the White House put a lot of pressure on,” said Sen. Angus S. King Jr., Maine independent. “Clearly, that made it harder.”

He and Sen. Susan M. Collins, a Republican who worked on the proposal, dismissed the administration’s complaints. While acknowledging they goofed on the future amnesty, they defended their move to place current illegal immigrants on low priority for deportation.

Ms. Collins said the goal was to let future illegal immigrants know that they will be priorities for ouster.

But they have been priorities dating back to the Obama administration, which said anyone who came in 2014 or later should be targeted for deportation.

Sen. James Lankford, an Oklahoma Republican who had been working with the group but dropped out after he saw the direction they were going, said those who wrote the proposal intended to protect the parents of Dreamers.

“The group determined they were afraid that President Trump would immediately deport all the parents once the DACA kids became citizens. And so their answer was to give amnesty to every adult,” Mr. Lankford said.

He said Democrats were so opposed to working with Mr. Trump that they rejected what should have been a highly praised bipartisan idea.

Mr. Lankford ended up supporting a Republican-led bill that enshrined Mr. Trump’s plan: legal status for 1.8 million people; $25 billion for border security along with changes to speed up deportations and change the calculus for future would-be illegal immigrants; limiting the chain of family migration to spouses and children; and ending the visa lottery.

That plan fared the worst of any of the three options Thursday, garnering just 39 votes. Most Democrats voted against it and were joined by some Republicans who said it was too harsh. A handful of other Republicans voted against it for being too generous with the amnesty for 1.8 million people.

“I find myself flabbergasted about where my own party is in this debate,” said Sen. Ted Cruz, a Texas Republican who voted against the Trump plan.

Thursday marked the fifth time in 12 years that the Senate has attempted major immigration reform. Only once has a bill been sent out of the Senate — in 2006 — and that plan never saw action in the more conservative House. A bill in 2013 won majority Senate support, but Democratic leaders never even sent it to the House.

It’s doubtful that the Senate’s bipartisan deal this year would have fared any better in the House this time, either.

Instead, Republican leaders in the lower chamber were trying to build support for their conservative bill.

A White House official said Thursday that they were “very close to having enough votes,” but Rep. Mario Diaz-Balart, Florida Republican, said he heard the whip count had gone poorly.

“I heard not well,” he said. “I didn’t see it, but I heard not well at all.”

Still struggling for a solution, some senators said they may try a “skinny” amnesty that would keep DACA-style protections in place for Dreamers, coupled with a small down payment on the border wall.

The White House has said it isn’t eager to accept such a “Band-Aid” solution.

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