WASHINGTON (Reuters) - Urgent negotiations aimed at shielding young, undocumented immigrants from deportation intensified on Thursday as Republican U.S. senators emerged from a meeting with President Donald Trump expressing confidence a deal could be struck this month.

Flanked by Senator Chuck Grassley (L) and Homeland Security Secretary Kirstjen Nielsen (R), U.S. President Donald Trump speaks during a meeting with Republican members of the Senate about immigration at the White House in Washington, U.S., January 4, 2018. REUTERS/Kevin Lamarque

As a follow-up to the Republican-only talks on so-called “Dreamer” immigrants, Trump is inviting senators from both major parties to the White House next week. Some senators told reporters the aim was to hold the meeting on Tuesday.

“Next week the president is inviting a bipartisan group of senators to the White House to discuss the next steps on responsible immigration reform,” White House spokeswoman Sarah Sanders told reporters.

For at least the past four months, Democratic and Republican senators have been meeting behind closed doors to try to fashion legislation that would protect at least 700,000 undocumented immigrants who were brought into the United States as children.

“I think we’re narrowing the differences,” Senator John Cornyn, the No. 2 Senate Republican, told reporters.

Even with evidence of progress, there were difficulties ahead.

Republicans continued to push for reigning in “chain migration” that could jeopardize the parents of Dreamers who are still in the United States illegally. And whatever deal that is cobbled together in the Senate could face opposition from hard-liners in the House of Representatives.

Some House Republicans have been clamoring to use Dreamer legislation to attach additional money for immigration enforcement throughout the country, which immigration advocacy groups fear would be used to go after the relatives of the young immigrants.

Many of these Dreamers are from Mexico and Central America and have spent most of their lives in the United States, attending school and participating in American society despite their illegal status.

Trump put their fates in doubt in early September when he announced he was ending Democratic former President Barack Obama’s Deferred Action for Childhood Arrivals, or DACA, program, which allowed the youth to legally live and work in the United States temporarily.

Unless the Republican-controlled Congress and Trump can agree on a legislative replacement, these immigrants will be subject to deportation at the beginning of March. For many, their protections already have expired.

Senator Thom Tillis, one of a handful of Republican senators who met with Trump on Thursday, returned to Capitol Hill saying it was possible a bill could hit the Senate floor later this month.

“I think so. I think if people get serious about it and start focusing and get in the room together. ... I think we’ll see progress next week,” Tillis told reporters.

Republican Senator James Lankford expressed hope that at next week’s bipartisan meeting Trump will produce a document that specifically lays out his administration’s demands for additional border security measures and other immigration law changes to accompany the new protections for Dreamers.

Trump and congressional Republicans have insisted on tougher border security, especially along the southwestern border with Mexico. Senate Democratic leader Chuck Schumer reiterated that Democrats also want “strong and real border security.”

Lankford said that a $1.6 billion injection of additional border security funds for next year was under discussion - a number that Democrats have embraced for more electronic surveillance, drone flights and other high-tech enhancements.

They have steadfastly opposed Trump’s call for building a southwestern border wall.