Washington (CNN) An effort to force a House vote on immigration didn't pick up any new supporters Tuesday night, but its backers say they are already sure it will reach enough signatures to hit the floor.

"We are extremely confident we already have the votes," Republican Rep. Jeff Denham of California said as he walked onto the House floor for the first votes of the week, which was the first opportunity lawmakers had to sign the measure since last week.

He walked into the Capitol with Republican Rep. Carlos Curbelo of Florida, who filed the so-called discharge petition on Denham's rule, which brings a floor vote on the Deferred Action for Childhood Arrivals policy. DACA protected young undocumented immigrants who came to the US as children, but President Donald Trump has decided to end it, though courts have temporarily paused that plan.

The two lawmakers are leading the charge among a group of moderate Republicans who are bucking their party leadership to push forward the petition , which circumvents leadership and the committee process.

If the petition can pick up 25 Republican signatures and those of every Democrat in the House, leadership would be forced to call four bills to the floor that address DACA. It currently has support from 18 Republicans and one Democrat , who signed earlier than the rest of her party last week because she expected to be out all of this week. The petition's backers still expect to hit the number of signatures this week.

Denham's rule would provide for debate and votes on four different immigration-related bills. One would be a bipartisan compromise, one would be a hardline bill supported by conservatives, one would be a Democratic bill to authorize just a version of the DACA program into law and one is completely up to House Speaker Paul Ryan -- leaving him free to choose any bill.

Leadership, however, is whipping against the measure , asking moderates to not sign it and emphasizing the importance of House Republicans keeping control of legislation and solving the problem on their terms, according to a Republican leadership aide.

On Tuesday, House Speaker Paul Ryan, R-Wisconsin, and Majority Leader Kevin McCarthy, R-California, traveled to the White House "to continue the conversation about addressing our broken immigration system," Ryan's spokeswoman AshLee Strong said in a statement.

"Obviously -- how do I say this kindly -- the speaker is not too happy," said Republican Rep. Mario Diaz-Balart of Florida, who has signed the petition. "I thought this was done in a way that was being respectful, but I understand that leadership is unhappy. ... But the reality is this is an issue that has to get solved."

Leadership may be successful in buying itself some time, but not indefinitely. Moderate Washington state Rep. Dan Newhouse, a Republican who has been a supporter of DACA compromise efforts, said he was still considering signing and has spoken with leadership about what else they may have planned.

"If they're not going to come forward with any alternatives, then unfortunately the discharge is one of the few options we have left," Newhouse said. "There's a clock that's ticking in order to get this done."

Denham said he'd been in contact with Republican leadership "several times" just since returning from the weekend, and that while he didn't know what they discussed at the White House, he knows that Trump is now "focused on" the issue along with leadership. As they walked in together, Curbelo said they were "not at all" concerned about the whipping against them.

"I'd hope they invest that time instead in finding a solution," Curbelo said, with Denham chiming in in unison on the last three words.

In a quirk of the "queen-of-the-hill" rule, the bill with the most votes advances to a final vote no matter how many of the bills get a majority of the House, leaving conservatives concerned that the process is set up to favor the bipartisan bill.

Conservatives are also watching very closely, according to a senior GOP aide, to see how leadership handles the insurrection. If moderates are handled with "kid gloves," the aide said, it could bode ill for members of leadership who are vying to replace Ryan when he retires and would empower the party's right flank to file its own similar rebellious measures.

"The responses are a little milquetoast compared to what conservatives usually get," the aide said. "The double standard is a little concerning."