(CNN) A Republican lawmaker-led effort to buck party leadership and force an immigration vote in the House picked up its first new GOP backer in a week Wednesday, but intense negotiations continued behind the scenes to find an alternative path forward.

Those negotiations also went another day without producing clear agreement among dueling Republican factions.

The civil war continues to put the squeeze on House Speaker Paul Ryan and his leadership team, who have worked overtime in the past week to convince their members not to allow the procedural move to succeed.

Minnesota Republican Rep. Erik Paulsen on Wednesday signed the petition to force a series of votes on immigration proposals, bringing the total number of House Republican members to sign to 21 and making him the first GOP congressman to sign since leaders and conservatives intensified their efforts to stave off the moderate Republican rebellion.

House GOP leaders, the moderates organizing the petition and top members of a conservative group of House Republicans who have staunchly opposed the effort all huddled in Ryan's office for more than an hour Wednesday afternoon, though, to continue negotiations about an alternative plan to have votes on the floor in June.

Petition organizer Rep. Jeff Denham, a California Republican, said Wednesday that he wouldn't push for the petition to gain enough signatures to force votes until those negotiations fall apart.

"We're not going to put all 25 across until we're done with that negotiation," Denham said.

Paulsen's signature brought the effort further within reach of succeeding. The signature means just five more Republicans are needed to force the issue -- assuming 192 of the 193 Democratic House members sign on. So far, only one Democrat -- Rep. Filemon Vela Jr. of Texas -- has said he will not sign the bill.

As of Wednesday evening, 184 Democrats had signed the petition. Vela had voiced for weeks that he had concerns about the petition being used a vehicle to get border wall funding but confirmed he would vote no to reporters Wednesday afternoon, saying "I'm not signing it," even if his signature were to be the deciding factor.

No path forward yet

All the members who gathered in Ryan's office left touting a good meeting without any clear details about what would happen next.

"Making good progress, nothing's decided at this point," said conservative Freedom Caucus Chairman Mark Meadows of North Carolina as he left.

"Certainly not a waste of time," said Florida Rep. Carlos Curbelo, who filed the petition. But, he added, the clock is ticking. "I think it's important to have a fairly specific agreement by the end of this week."

Meadows said the group had "more than a reasonable chance" of reaching agreement by Thursday, when the House is scheduled to wrap up business for a weeklong recess. The petition must be physically signed in person.

"I hope so, we're getting pretty close," Denham said when asked if it could happen.

Leadership also scheduled a full GOP conference meeting on the issue of immigration, slated to last two hours, on June 7.

Paulsen's signature Wednesday morning came despite GOP leadership working hard to convince Republicans to hold off. In a statement, Paulsen said leadership had run out of time.

"President Trump and leaders in Congress promised to fix DACA. They haven't," Paulsen said, referring to the Deferred Action for Childhood Arrivals program, which protected young undocumented immigrants who came to the US as children and which President Donald Trump opted to end last year. "I took action to solve this problem and there's bipartisan support to do so."

Before Paulsen's signature, there had been little movement from Republicans this week. Members and aides have said that that is partially because moderates tend to believe leaders when they say that they are working on a solution. Some members still want to give leadership the benefit of the doubt, waiting to see if Ryan and others can pull a solution together.

Many members behind the petition want an answer before heading home for the weeklong recess, though the petition could still force votes on June 25 if it is signed the first week that lawmakers return in June.

Sticking points

One of the emerging issues is what happens to the undocumented immigrants themselves.

Moderates have insisted on a pathway for those young people to become citizens, after a lengthy waiting period and only if they meet certain criteria. Curbelo called that permanent solution "fundamental" to a deal.

But conservatives call such a plan a "special pathway" -- opposing anything that grants a way for an undocumented immigrant who can't otherwise become a citizen without leaving the country for at least a decade, as is the case for the vast majority of DACA recipients.

Moderates have pushed for a bipartisan bill that includes a pathway to citizenship and border security, but leadership has thus far said they don't believe that bill could pass the Senate and be signed by the President so it won't move forward.

Leadership has already guaranteed a vote to conservatives on a bill they are supporting that isn't likely to pass the House, but hasn't specific what moderate bill will get called with it.

Conservatives are also making clear that they expect anything Ryan might call alongside their bill meet an informal Republican conference rule that requires any bill on the floor to have a majority of Republican Party votes, which the bipartisan bill likely does not.