One Democratic senator, Jon Tester of Montana, told reporters that negotiators on the bicameral, bipartisan committee could reach a deal as soon as today.
"[It's] entirely possible we could have a deal in a timely manner, which could be tomorrow but certainly by the weekend," Tester said on Thursday.
House and Senate negotiators are hoping to hammer out a deal before Monday, which would leave them with just enough time next week to vote on the measure in both chambers before the government would shut down at midnight on Feb. 15.
"I don't know that he'd sign whatever we came up with, but I had a very positive conversation with the president and the vice president," Shelby said. "I believe he was very reasonable with us."
Trump indicated to Shelby that he's "looking to keeping the government open" and wants to find a legislative solution to building his prized border wall.
"We'll see what happens, but I certainly hear they're working on something, and both sides are moving along," Trump told reporters at the White House following his meeting with Shelby. "We need border security. We have to have it. It's not an option."
Shelby, for his part, admitted it's not an easy task.
"It's going to take a lot of money and a lot of years, a lot of building and a lot of technology, and a lot of manpower," Shelby said. "But I think it's important."
Negotiators are discussing a deal that would fund the government for the current fiscal year, Shelby told reporters.
But it's unclear how much funding, if any, would go to the border wall.
His comments came after Senate Majority Leader Mitch McConnell told reporters earlier on Thursday that he's not interested in another short-term spending deal like the one passed late last month.
Another top negotiator on the committee, Sen. Roy Blunt, R-Mo., told reporters on Thursday that committee members, including Democrats, aren't entirely opposed to building barriers.
"A lot of this is extending walls that other presidents have built, and improving walls other presidents have built," Blunt said.
House Speaker Nancy Pelosi also expressed optimism -- if only Trump would lay low -- that the government won't shut down again in a week, expressing confidence that appropriators would strike a deal "to protect our borders as we protect our values."
"I have confidence in the appropriators not because I know what they are doing, but because I have confidence in the appropriations process," Pelosi said earlier Thursday, emphasizing she would support a "fair conclusion."
"I ask the administration to be as non-interventionist as I am on that," she added. "Just let them do their work, and hopefully we'll get some good news in a short period of time and certainly in time for the deadline of February 15."
ABC's John Parkinson and Sarah Kolinovsky contributed to this report.