"It doesn't work," Durbin said.
Immigration issues have been core to that debate, including the president's proposed border wall and Democrats' call for protections for so-called 'Dreamers,' the approximately 800,000 young immigrants who were brought illegally to the U.S. as children.
"It is true that Chuck Schumer made what I considered to be a bold and important concession and said, 'Yes, we'll go forward with the wall as long as we do this in a responsible fashion,'" Durbin responded.
Stephanopoulos asked if it was correct that the border wall would cost $20 billion.
"I'm not going to quote numbers because I don't think that's my place ... but I can tell you it was a substantial commitment to the president. The president embraced it. And Chuck came back to the Hill."
But, Durbin added, "Two hours later, a call from the White House says, 'The deal is off. We're not going to stand by this at all.'"
The senator added, "How can you negotiate with the president under those circumstances where he agrees face-to-face to move forward with a certain path and then within two hours calls back and pulls the plug?"
Asked if Congress will reach a deal in time for the government to reopen Monday, Durbin said he couldn't answer but added, "There are bipartisan conversations going on right now."
In a separate appearance on "This Week" Sunday, North Carolina Republican Congressman Mark Meadows disputed the notion that the president had reached an agreement with Schumer.
“The president is the president of the United States,” the congressman said. “He makes his own decisions.”
“Any suggestion that I'm going to convince the president to go one way or another greatly exaggerates my influence," Meadows said. "And I will say this, there was no deal. There is no deal.”
Meadows also took issue with reports about how the border wall would be funded. "You were talking about Schumer offering $20 billion over seven years -- he can't do that," Meadows said.
Meadows continued, “You can't obligate a future Congress. So even if [Schumer] made that offer, he can't even deliver on the very offer that he's putting forth. So when we look at this, let's call it what it is. This is a tactic.”