Washington (CNN) As the dust settled Monday on an agreement to reopen the government, the path forward for immigration remained as murky as ever.

Democrats and Republicans who worked to break the impasse over the shutdown spun their vote to accept a slightly shorter continuing resolution as a victory because of a commitment to turn to immigration. But the future of the Deferred Action for Childhood Arrivals policy and discussions on border security are undetermined.

"Well, there's conversations already started, bipartisan conversation, about whether we can come up with a bipartisan Senate bill before February 8," said Senate No. 2 Democrat Dick Durbin, who had been pursuing a DACA compromise for months.

The "hope," he said, for those who pushed for a promise to move to immigration is that if a bill can pass the Senate with a strong bipartisan vote, President Donald Trump may endorse it and push the House to act.

Since Trump ended DACA , which protects young undocumented immigrants who came to the US as children, lawmakers have worked to find a way preserve the popular program while meeting the President's and Republicans' demands for border security and immigration enforcement changes along with it.

The White House on Monday continued to meet with Republican senators, many of whom are conservative hardliners, as it has remained opposed to bipartisan proposals that have been floated thus far.

Still, Senate Majority Leader Mitch McConnell pledged Monday to consider an immigration bill, including DACA, sometime soon.

"it would be my intention to take up legislation here in the Senate that would address DACA, border security, and, related issues as well as disaster relief, defense funding, health care, and other important matters," McConnell said Monday, saying the process would have "a level playing field" and be "fair to all sides."

After a brief weekend shutdown, Congress on Monday voted to fund government until February 8 -- which will be the new deadline for any agreement between the parties on immigration and other outstanding issues. Absent agreement, McConnell said, the Senate will move to an open debate.

That was enough to convince a number of Democrats to support the funding bill -- but they all indicated they expected to see the promise delivered.

"Trust but verify is my motto," said Maine Sen. Angus King, an independent who caucuses with Democrats. "He's made this commitment publicly, he made it on the floor of the Senate. ... I think this is an important opportunity for him to demonstrate that he will carry through."

A 'Gang of 60' or more?

Bipartisan talks will continue -- but lawmakers were expected to broaden beyond the core group of negotiators who had hammered out a compromise previously.

The founders of the Gang of Six senators that had brought a bipartisan proposal to Trump, only to have it crudely rejected , said that original group will no longer be operative as supporters of its work aim to get something that can pass the full Senate, where 60 votes are needed to advance legislation and Republicans have only a 51-49 majority.

"The Gang of Six started the process," Graham said. "That's all it was there to do. We need the Gang of 60. So the Gang of Six is going to be replaced by the Gang of 60."

"It's a new gang," Durbin told CNN. "Some of the old, some of the new."

And responding to Graham's assessment of a "Gang of 60," Durbin said it may need to be even bigger.

"Maybe 70, I don't know," Durbin said. "We need, if we can, to find a path to get this done."

The White House and Republican leadership has been pressing for a group of the congressional "No. 2's" -- the seconds in command in each party in each chamber -- to be the main vehicle for negotiations.

White House chief of staff John Kelly and at times Homeland Security Secretary Kirstjen Nielsen have also been participating in those talks.

But further meetings haven't been scheduled since the government funding votes last week, aides said, and Democrats have long been skeptical the talks are just designed to slow things down.

"I don't think that group was ever intended to work, it was intended to push things past the deadline," said a Democratic senator, speaking anonymously to be candid. "It was intended to slow-roll the work."

What about the GOP House?

Even with the Senate commitment, there was no such indication from the House -- leaving senators relying on hope that the lower chamber could follow suit should senators pass a bill.

One influential House conservative, Freedom Caucus Chairman Mark Meadows, said the House should pass something "as conservative as it possibly can be" and then send it back to the Senate.

"Hopefully (then) go to conference and find a compromise that we can send to the President's desk that represents the will of the people," Meadows told CNN. "But it needs to start here -- it can't start in the Senate."

Immigration advocacy groups and supporters immediately criticized Democrats for giving up too easily.

"This simply kicks the can down the road with no assurance that we will protect Dreamers from deportation or fight Republican attempts to curtail or eliminate legal immigration," said Illinois Democrat Rep. Luis Gutierrez, a longtime immigration advocate, in a statement. "I do not see how a vague promise from the Senate Majority Leader about a vague policy to be voted on in the future helps the Dreamers or maximizes leverage."

White House role

And immediately after the vote to break the impasse on funding, six Republican senators including several hardliners traveled to the White House to meet on the Issue with Trump.

They said the White House wanted their ideas on the four areas the President has identified as his priorities for this deal -- DACA, border security, cutting family-based migration and ending the diversity visa lottery.

"We were just talking about all the issues the President identified," Senate Majority Whip John Cornyn said of the meeting. "Different ideas about how to address them in creative ways."

The White House did invite two Democrats to meet with the President Monday afternoon, Sens. Joe Manchin and Doug Jones. Both come from deeply red states and have not been involved in immigration policy discussions.

One Republican who was part of the Gang of Six, Arizona Sen. Jeff Flake, expressed frustration with any partisan conversations.

"I'm not doing anymore immigration negotiations with just Republicans, that's fruitless," Flake told CNN on Saturday. He repeatedly criticized the President for changing his mind, and had pushed McConnell in multiple meetings over the weekend to commit to move regardless of Trump's approval at the time.

"if we can get an agreement with the White House, that's great. I'm not holding my breath," he added.

Asked Monday what his advice for Trump was, Graham implored the President to refrain from blowing up talks.

"Be constructive, just be constructive," Graham said.