Some Democratic lawmakers are pushing back on including changes to "chain migration," which allows citizens and permanent residents to sponsor family members, as well as changes to the State Department's Diversity Visa Lottery system as part of the potential deal.

The pushback underscores the uphill climb negotiators face in cobbling together an agreement that satisfies either party's base.

Members of the Congressional Tri-Caucus — which includes the Asian Pacific American, Black and Hispanic caucuses — said after a meeting with Democratic leadership Tuesday night that they did not believe changes to family-based immigration or the diversity visa lottery should be part of any agreement.

Rep. Linda Sánchez (D-Calif.) said she has "real issues" with linking either issues to a fix for the Deferred Action for Childhood Arrivals (DACA) program.

"I even have issues with the border security element of DACA and dreamers. ... If we want to talk about border security, you know, let's do it in a comprehensive immigration bill," she said.

Lawmakers said after a White House meeting that they had agreed to link the two issues , as well as border security, to a fix for DACA.

Durbin, who has taken the lead on negotiating for Senate Democrats, acknowledged that rank-and-file members had concerns.

"There's so much uncertainty. We don't have anything really nailed down. People are worried about the direction," he told reporters.

Asked if he would support the four parameters being outlined, he added that it would come down to the details of what was ultimately included.

"I'm just saying that there are ways to do things that are painless and ways that are fatal," he said.

Hoyer also downplayed that Democrats had committed to a framework that linked family-based immigration reforms, DACA, border security and the visa lottery.

"I'm prepared to discuss resolving the issue of the Dreamers. ... Now that does not mean I am for X, Y or Z," he said.

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

Durbin, however, said Tuesday night that Democrats are sticking with their demand while acknowledging the short timeframe.

"It's even worse than you think. It's 10 days away by a generous count and four of those days we're not in session," he said.

Asked about McConnell's comments, Durbin declined to speculate on whether or not more Democrats would vote "no" on an additional continuing resolution if there isn't a DACA deal.