The Democratic members of the Senate’s immigration “gang of eight” headed down Pennsylvania Avenue for a White House meeting, but public work on their bill — and the key issue of border security — will wait until next week.

In the meantime, negotiations are under way to try to thread the needle on border security, to come up with a specific plan that will win Republican votes and maintain Democratic support.

“We are trying to set achievable goals that [Democrats] can support,” Sen. John McCain of Arizona, a GOP member of the gang of eight, told reporters.

McCain explained that what’s under discussion would be somewhat different from a border security amendment already filed by Minority Whip John Cornyn, R-Texas.

The amendment “would define ways that can assure them that we can achieve 90 percent effective operational control of the border, that we have sufficient funds devoted to it, and that we have sufficient other measures taken to ensure that we have border security and still keep in the parameters of what the Democrats can support,” he said.

Asked about the border security work, fellow gang of eight senator Marco Rubio, R-Fla., noted that Cornyn wasn’t the only one with an enhanced border security amendment.

“There are others working on a border plan as well. We’ll figure out which is the best one,” Rubio said. “I don’t think border security will kill the underlying bill.”

Rubio conceded some pieces of the Cornyn amendment cause problems for the underlying bill, saying that he, too, wanted to see a specific border plan before Senate passage.

“There might be some elements of it that can’t get the votes necessary to pass, but I think — he’s doing it in good faith. He’s working very hard, he lives on the border, he’s a member of our leadership,” Rubio said.

Slow Legislative Process

While senators took their first vote on a proposed amendment to the immigration package Thursday morning, the real work is waiting for next week. Senate Majority Leader Harry Reid, D-Nev., is even telling senators to be prepared for work next weekend.

The first vote came on a Reid motion to kill an amendment from Sen. Charles E. Grassley, R-Iowa, that would have blocked issuing provisional immigrant status until there’s been six months of border control. Sen. Lisa Murkowski, R-Alaska, joined the gang of eight’s Republicans in supporting Reid on that vote, a sign her vote might be attainable.

“We just began,” Murkowski said when asked if she could support the bill going forward. “I think that what the eight group put together is … a good start.”

Speaking with reporters, Rubio explained his opposition to the Grassley proposal.

“People don’t realize this, but those folks when they come forward, all they get is a temporary status and a work permit while they’re in that temporary status,” Rubio said. “They don’t qualify for any benefits or even citizenship for that matter, but they have to pay a fine, and that fine money is what we’re going to use to pay for the border security. If we don’t get that fine money, it’s going to fall on the American taxpayer, and I want to avoid that.”

Still, opponents of the gang of eight proposal renewed concerns about the process that Reid’s employing on the floor.

“I am concerned, and Sen. Grassley is concerned, about making this a 60-vote, every amendment … a 60-vote margin,” Sessions said. “Normally, you don’t have every vote 60 votes, and you could have something big that might — would. So, I’m not sure where Sen. Grassley is with Sen. Reid at this point.”

Sessions added that he would oppose an effort to have a “vote-a-rama”-style batch of amendments come up for votes as next weekend approaches, with a long sequence of amendments dispensed with all at one time. Because the Senate uses complex unanimous consent agreements to do its work, such vote sequences have become fairly common on big-ticket legislation.

He added that he would continue to push for amendments of his own.

“People want to vote for something. I mean, that makes you feel good,” Sessions said. “You want to fix the thing, but we — we’ve been hijacked by legislation that won’t work.”

Humberto Sanchez contributed to this report.