President Trump is sending ICE agents to airports starting Monday to help relieve security lines despite sharp criticism from Democrats and the union representing Transportation Security Administration workers.

Democrats said posting U.S. Immigration and Customs Enforcement agents at airports without training could lead to more chaos and potentially violence.

Some Republicans said they were “not a fan” of the move, as Sen. Lisa Murkowski of Alaska put it.

Senate Majority Leader John Thune, South Dakota Republican, called the ICE deployment a tool of “last resort” stemming from Democrats’ refusal to fund the Department of Homeland Security despite some White House concessions on immigration enforcement demands.

“If the Democrats do not allow for Just and Proper Security at our Airports, and elsewhere throughout our Country, ICE will do the job far better than ever done before!” Mr. Trump said on social media. “The Fascist Democrats will never protect America, but the Republicans will.”

White House border czar Tom Homan confirmed that the deployment would begin Monday. He said Mr. Trump called him Sunday morning and made clear he had run out of patience.

“He’s tired of waiting, the American people being held hostage at the airport, TSA not being paid, FEMA — all these people, federal employees, are working for free,” Mr. Homan said on “Fox News Sunday.”

Close to 400 TSA officers have quit since the department shutdown began, and some airports are missing 30% to 40% of their staff on a given day. Travelers have faced waits of two to three hours at major airports, and spring break demand is compounding the pressure.

Mr. Homan said ICE agents would take on tasks that don’t require specialized TSA training, such as guarding exit lanes and checking identification, freeing TSA officers to focus on X-ray screening and other work that demands their expertise.

“We will put together a plan today, and we will execute tomorrow,” he said on CNN’s “State of the Union.”

Senate Minority Leader Charles E. Schumer, New York Democrat, called the deployment of ICE agents to airports “disturbing” and another “impulsive” idea Mr. Trump has floated that his administration must hastily implement.

“ICE agents who are untrained and have caused problems everywhere they’ve gone, lurking at our airports — that’s asking for trouble,” Mr. Schumer said Sunday on the Senate floor.

“It will certainly make the chaos at the airports even worse. No one has any faith in ICE agents,” he said.

Mr. Schumer said Mr. Trump should instead persuade Senate Republicans to support Democrats’ efforts to fund TSA separately from other parts of the Homeland Security Department.

Republicans have rejected Democrats’ piecemeal efforts to fund the department’s agencies as a political ploy to defund law enforcement, but they have begun discussing new strategies they could deploy if Democratic intransigence continues.

“There’s a number of different ideas, and I honestly couldn’t tell you which one we’re going to land on at this point,” Sen. John Hoeven, North Dakota Republican, told The Washington Times.

Democrats have said they will not fund the entire department without immigration enforcement policy changes that they say the American people are demanding after federal agents killed two U.S. citizens in Minneapolis this year.

They also want independent investigations into those shootings and accountability for any findings of wrongdoing.

“The FBI is still not sharing information about the investigations into Alex Pretti or Renee Good with state and local officials,” Sen. Chris Van Hollen, Maryland Democrat, told The Times. “How can we trust DHS to conduct an independent investigation when they called them domestic terrorists? So I don’t know of any progress on that.”

Democrats’ other demands include requiring immigration enforcement agents to stop wearing masks, avoiding “sensitive locations” such as schools, hospitals, courts and polling locations, and requiring judicial warrants for arrests on private property.

“It’s great that the White House is working with us, but we still have a ways to go when it comes to masks and search warrants and other demands,” said Sen. Catherine Cortez Masto of Nevada, one of the Democrats who has participated in bipartisan negotiations with the White House.

Mr. Homan said the two sides are having “good conversations” but that the administration will not surrender ICE’s congressionally mandated authorities.

He framed the dispute as a “policy execution issue” rather than a policy fight. He said the enforcement practices Democrats want to change are largely already in place.

The White House sent its latest counteroffer to Democrats on Friday evening and had not heard back as of Sunday afternoon.

“They were supposed to come with an offer [Saturday] night, and instead they canceled the meeting altogether,” said Sen. Susan M. Collins, Maine Republican and chair of the Appropriations Committee. “So I don’t know what’s going to happen.”

Mr. Thune warned that if a deal does not come together in the next couple of days, then “we’re going to be in a pretty bad place.”

He said the president’s deployment of ICE agents to airports is “evidence of how sort of desperate things have become at our airports.”

TSA agents are set to miss another paycheck at the end of the week if the funding issue is not resolved.

“I think you’re going to see more TSA agents as we come to Thursday, Friday, Saturday of next week. They’re going to quit, or they’re not going to show up,” Transportation Secretary Sean Duffy said on ABC’s “This Week.”

Democrats said posting ICE agents at airports will not help compensate for missing TSA agents.

“The last thing that the American people need are for untrained ICE agents to be deployed at airports all across the country, potentially to brutalize or, in some instances, kill them,” House Minority Leader Hakeem Jeffries, New York Democrat, said on CNN.

The American Federation of Government Employees, which represents more than 50,000 TSA officers, echoed that warning.

AFGE National President Everett Kelley said TSA officers spend months learning to detect explosives, weapons and threats specifically designed to evade detection. He said these skills cannot be improvised.

“Putting untrained personnel at security checkpoints does not fill a gap,” Mr. Kelley said. “It creates one.”

Mr. Kelley said TSA officers have worked without pay for more than five weeks and hundreds have quit. He called on Congress to act.

“Congress has the power to fund TSA today,” he said. “It’s time for them to stop playing politics and do their jobs.”