Senators working on a bipartisan immigration plan indicated Wednesday that they are closing in on an agreement, with a deal possible this week.

Sen. Dick Durbin (D-Ill.) told reporters that he hopes the group can reach an agreement before this weekend that would pair a fix for the Deferred Action for Childhood Arrivals (DACA) program with parameters agreed to after a White House meeting.

"We're close. The president made it clear what's important to him and we're trying to figure out how to do it in a thoughtful way and not just include fences and things of that nature, barriers, but to go beyond into technology, which the agency, people working there, over and over again said that's where we should start," he said.

They are still trying to iron out the details of what border security package would be included, but Durbin said the agreement would include a pathway to citizenship for immigrants brought into the country illegally as children who meet certain requirements.

In addition to border security and DACA, the bipartisan Senate group, which has been negotiating for months, has also discussed changes to the diversity lottery program - potentially shifting those visas toward Temporary Protected Status recipients - and changes to family-based immigration.

Sen. Jeff Flake (R-Ariz.), who is also part of the negotiations, echoed Durbin, separately on Wednesday telling NBC News that senators are "very close," with a deal potentially being announced this week.

Spokespeople for Flake didn't respond to a request for comment about the status of the negotiations.

Even if the bipartisan group - which also includes Sens. Bob Menendez (D-N.J.), Lindsey Graham (R-S.C.), and Michael Bennet (D-Colo.)- is able to get an agreement amongst themselves they would still need to win over 60 senators, not to mention the more conservative House and President Trump.

on Wednesday

House conservatives outlined their own bill, which includes more aggressive enforcement measures.

GOP Sens. Tom Cotton (Ark.), Chuck Grassley (Iowa) and David Perdue (Ga.) noted that they also think a deal needs to include more enforcement "so that federal law officers can protect law-abiding American citizens and immigrants."

Durbin acknowledged that Cotton, who introduced a bill with Perdue earlier this year that would overhaul the legal immigration system, would be unlikely to support any deal his group comes up with.

"We're not going to win Tom Cotton, okay? ... And we're not going to get every Democrat either. Some of them, if we get to an agreement, are going to say it's not good enough," he said.

Senate Majority Leader Mitch McConnell (R-Ky.) has committed to giving a DACA bill a vote if Trump indicates he would be willing to sign the legislation. Trump reiterated his call for a "wall" on Wednesday, saying it "must be part of any DACA approval."

Lawmakers, following Tuesday's White House meeting, have suggested the U.S.-Mexico border wall could be a fence and wouldn't need to cover the entire border.

Meanwhile, Durbin, Sen. John Cornyn (R-Texas) and Reps. Kevin McCarthy (R-Calif.) and Steny Hoyer (D-Md.) met on Wednesday to begin negotiating a timeline for DACA legislation.

But both parties appear to be operating under different deadlines, with Republicans pointing to March as their cutoff date, but Democrats adamant they want a deal by next week's government funding deadline.

Durbin noted that the four lawmakers agreed to have a staff meeting on Thursday "to discuss an agenda" but appeared skeptical that they would be able to come up with a deal by Jan. 19, the deadline to fund the government.

The Trump administration announced last year that it is ending the DACA program, which allows certain immigrants brought into the country as children to work and go to school.

That decision gave Congress until March to codify the program, though a court threw a curveball into that timeline on Tuesday by ruling that the Trump administration has to keep DACA in place while litigation plays out.