* Schumer on Trump: "like negotiating with Jell-O"
* Trump: "Democrats wanted to give me a nice present" (Adds quotes, impact of shutdown)
WASHINGTON, Jan 20 (Reuters) - President Donald Trump and Republican lawmakers took a tough stance on Saturday after the U.S. Congress failed to fund federal agencies, saying they would not negotiate on immigration until Democrats help end the government shutdown.
Funding for federal agencies ran out at midnight with no agreement in Congress, meaning the second year of Trump's presidency began without a fully functioning government. Lawmakers failed to resolve an impasse over Democrats' demands that any short-term spending legislation include protections for young undocumented immigrants.
U.S. government workers were told to stay home or, in some cases, work without pay until new funding is approved in the first federal government shutdown since a 16-day funding lapse in October 2013.
A trip by Trump and some Cabinet members to the World Economic Forum in Davos, Switzerland, was in flux, White House budget director Mick Mulvaney said, with the situation being assessed on a day to day basis.
Parks, open-air monuments and Smithsonian museums were open in the U.S. capital as a women's rights march took place on the National Mall. But visitors were turned away from the Liberty Bell in Philadelphia and the Statue of Liberty in New York Harbor.
The Republican-controlled Senate and House of Representatives held rare weekend sessions on Saturday, facing a political crisis that could have an impact on congressional elections in November.
Both Republicans and Democrats dug in, with each side blaming the other. Republicans said they would refuse to negotiate on immigration until Democrats provide the votes to re-open the government. Democrats insisted they have been willing to compromise but Republicans backed out of deals.
"The president will not negotiate on immigration reform until Democrats stop playing games and reopen the government," said White House spokeswoman Sarah Sanders.
Marc Short, the White House's legislative affairs director, said Trump had been in contact with Republican leaders in Congress during the day, but had not reached out to Democrats.
Short said the president likely would be most effective making the case for ending the shutdown directly to the American people, and he did not rule out Trump addressing the nation in the coming days.
The tough message from the White House and Republicans in Congress led to speculation that Washington could be in for a prolonged political battle.
Speaking at the U.S. Capitol, Senate Democratic Leader Chuck Schumer delivered a stinging portrayal of Trump as an unreliable negotiating partner, saying the two sides came close to an agreement several times only to have Trump back out at the urging of anti-immigration conservatives.
"Negotiating with President Trump is like negotiating with Jell-O," said Schumer, who met Trump at the White House on Friday for a 90-minute meeting that had briefly raised hopes. "It's impossible to negotiate with a constantly moving target."
The federal government had been running on three consecutive temporary funding bills since the new fiscal year began in October.
Democrats had sought to secure permanent legal protections for 700,000 young undocumented immigrants as a condition for new government funding after their attempts to push through the protections in stand-alone bills were rebuffed.
Senate Republican Leader Mitch McConnell said a solution to the crisis was "just inches away" but he blamed Democrats for blocking legislation to pass the fourth stopgap funding measure.
Lawmakers held informal discussions at the Capitol on Saturday as senators debated the shutdown on the Senate floor.
One idea floated by Republicans was to renew government funding through Feb. 8 to end the shutdown, while working to resolve other issues, including immigration, military and non-military spending, disaster relief and some healthcare matters.
Republican Senator Jeff Flake said a vote could be held either Saturday or Sunday on such a three-week spending bill. Republicans hoped Democrats would support it if they were assured McConnell would allow several immigration measures to come up for a vote. Under the strategy, whichever measure passed the Senate could gain Trump's support and pass the House.
The partial government shutdown was triggered at midnight on Friday when the Senate failed to agree to a House-passed bill to fund the government through Feb. 16. The bill drew strong opposition from Democrats and some Republicans.
Democrats and many Republicans want permanent legal status leading to citizenship for so-called Dreamers, immigrants brought into the country illegally when they were children. Trump ordered the Deferred Action for Childhood Arrival program to expire in March, requiring Congress to act.
The shutdown began a year to the day after Trump was sworn in as president. He portrayed himself as the ultimate dealmaker but his inability to cut a deal despite having a Republican majority in both houses of Congress marked arguably the most debilitating setback for his administration.
"This is the One Year Anniversary of my Presidency and the Democrats wanted to give me a nice present," he said on Twitter.
"Democrats are far more concerned with illegal immigrants than they are with our great military or safety at our dangerous southern border," he said. "They could have easily made a deal but decided to play shutdown politics instead."
Trump said the shutdown showed the need to win more Republican seats in 2018 congressional elections.
The immediate impact of the government shutdown was eased somewhat by its timing, starting on a weekend when most government employees normally do not work.
The Defense Department said its combat operations in Afghanistan and other military activities would continue, while federal law enforcement officers also would remain on duty.
The U.S. Trade Representative staff will continue talks on the North American Free Trade Agreement, major cybersecurity functions will continue, and most of the Environmental Protection Agency will remain open, budget director Mulvaney said.
But without a quick deal, hundreds of thousands of government employees will be put on temporary unpaid leave.
"It's ironic that they get paid - meaning Congress - and the rest of the government doesn't," said Dawn Gaither, 57, a Washington teacher. "That's what we need to do, kick these guys in the tail and get them to work."
(Reporting by Richard Cowan, Ginger Gibson, James Oliphant, Ian Simpson, and David Brunnstrom in Washington; Writing by Richard Cowan and Doina Chiacu; Editing by Bill Trott and Daniel Wallis)