By Tal Kopan, CNN

(CNN) -- One way or another, House Republicans have to find a way forward on immigration this week.

Tuesday, they're returning from Memorial Day recess away from Washington lacking substantive progress on inter-party negotiations and with a deadline at the end of the week on a way to resolve the roiling civil war within Republican ranks.

And the effort to force the immigration issue to the House floor got a boost Tuesday -- when a Democrat who previously said he would not sign the procedural move that would do so, which would have required Republicans to put up an extra signature, changed course.

The feuding between moderates and conservatives within the party has put GOP leadership in a no-win situation, either infuriating one side of their conference or being rendered spectators to the success of a procedural maneuver spearheaded by their moderate colleagues to take the power out of their hands entirely.

Leadership-spurred talks between moderates and conservatives have continued over the break, according to sources close to the negotiations, but no major breakthroughs have happened. A major sticking point remains a disagreement over a solution for young undocumented immigrants currently in limbo.

Florida Rep. Carlos Curbelo, one of the moderate Republicans organizing the effort, described "good talks" over break with conservative members, but didn't speak of any breakthroughs. Other sources described the situation similarly, with no movement on the key issues and a need for members to get back together in DC to make any final decisions.

The issue will come to a head by Thursday morning -- when leadership has scheduled an all-member meeting of the Republican conference to last two hours.

In the meantime, Tuesday will kick off a series of last-ditch meetings and closed-door negotiations to try to chart a path out of a seemingly irreconcilable divide.

At issue is what's called a discharge petition, a rarely successful House procedure that allows a bill to be called straight to the floor without leadership's or the majority's approval. A group of House Republican moderates have organized such a petition, and it stands five votes away from success, needing at least two more Republicans.

While Texas Democrat Rep. Filemon Vela had previously said he would not sign the petition regardless of whether his signature was the deciding factor on it succeeding, in a statement Tuesday he reversed course and said after meeting with constituents and clergy over break he would sign the petition.

Angry conservatives who oppose the tactic -- which would call to the floor four competing immigration bills but is designed in a way that will likely result in a bipartisan compromise bill winning out -- lashed out by tanking an unrelated agriculture bill, prompting leadership to convene talks among the factions.

Those talks have been more productive than any under this administration, the members have said, but have remained held up by a key issue: what to do about the end of the Deferred Action for Childhood Arrivals program, which protected young undocumented immigrants who came to the US as children.

The moderates who launched this effort are clear that they will not support any bill that does not have a way for those individuals to eventually legalize their status and apply for citizenship, a process that is almost impossible for the vast majority of DACA recipients under the status quo. The moderates' position is supported by virtually all Democrats, as well. Conservatives, meanwhile, have pushed for a hardline immigration bill that contains a host of aggressive anti-illegal immigration measures and cuts to legal immigration, which also would not offer DACA recipients anything more than renewable work permits, as they can get under the current program.

Neither of those positions have changed substantively over the course of talks, and before lawmakers left town, moderates essentially said if conservatives can't agree on a path to citizenship, further negotiations would be fruitless.

Leadership has promised conservatives a vote on their bill, which is not expected to have enough Republican votes for passage, although conservatives tanked the unrelated bill anyway. But moderates have not been promised any specific legislation for a vote, only that they will have something as well.

Moderates remain confident publicly and privately that they can get enough signatures to force their petition. That means if conservatives won't agree to a pathway to citizenship for DACA recipients, as is widely expected, they plan to proceed with their effort to force the votes. Based on procedural rules for a discharge petition, the last day they can get signatures and have the vote this month is Friday.

This story has been updated.